Subgroups — C₂×C₆₇×#2447 #30713949

Order 17152.

3070 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#124×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×D₄×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2447	C₂	1	16	yes	maximal, direct product factor
4	C₂×#397×C₆₇	C₂	1	2	yes	maximal, complemented
5	C₂×#397×C₆₇	C₂	1	2	yes	maximal, complemented
6	C₂×#391×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#391×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#390×C₆₇	C₂	1	1	yes	maximal, characteristic
9	C₂×#389×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#385×C₆₇	C₂	1	2	yes	maximal, complemented
11	C₂×#381×C₆₇	C₂	1	2	yes	maximal, complemented
12	C₂×C₄×C₄⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂³×Q₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₂³×D₄×C₆₇	C₂²	1	1	yes	characteristic
15	C₂⁴×C₄×C₆₇	C₂²	1	2	yes	—
16	C₂×#124×C₆₇	—	2	8	no	—
17	C₂×#124×C₆₇	C₂²	1	8	yes	—
18	C₂×C₄×D₄×C₆₇	—	2	8	no	—
19	C₂×C₄×D₄×C₆₇	C₂²	1	8	yes	—
20	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
21	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
22	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
23	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
24	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
25	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
26	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
27	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
28	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
29	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
30	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
31	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
32	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
33	#397×C₆₇	C₂²	1	16	yes	complemented
34	#397×C₆₇	C₂²	1	16	yes	complemented
35	#391×C₆₇	C₂²	1	8	yes	complemented
36	#391×C₆₇	C₂²	1	8	yes	complemented
37	#390×C₆₇	C₂²	1	8	yes	—
38	#389×C₆₇	C₂²	1	8	yes	complemented
39	#385×C₆₇	C₂²	1	16	yes	complemented
40	#381×C₆₇	C₂²	1	16	yes	complemented
41	C₄×C₄⋊C₄×C₆₇	C₂²	1	8	yes	complemented
42	C₂×#95×C₆₇	C₂²	1	1	yes	characteristic
43	C₂×#95×C₆₇	C₂²	1	1	yes	characteristic
44	C₂×#95×C₆₇	C₂²	1	2	yes	—
45	C₂×#95×C₆₇	C₂²	1	2	yes	—
46	C₂×#95×C₆₇	C₂²	1	2	yes	—
47	C₂×#95×C₆₇	C₂²	1	2	yes	—
48	C₂×#95×C₆₇	C₂²	1	2	yes	—
49	C₂⁵×C₆₇	C₂³	1	2	yes	—
50	C₂²×Q₈×C₆₇	—	2	4	no	—
51	C₂²×Q₈×C₆₇	—	2	4	no	—
52	C₂²×Q₈×C₆₇	—	2	4	no	—
53	C₂²×Q₈×C₆₇	C₂³	1	4	yes	—
54	C₂²×D₄×C₆₇	—	2	4	no	—
55	C₂²×D₄×C₆₇	—	2	4	no	—
56	C₂²×D₄×C₆₇	D₄	1	8	yes	—
57	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
58	C₂³×C₄×C₆₇	—	2	8	no	—
59	C₂³×C₄×C₆₇	—	2	8	no	—
60	C₂³×C₄×C₆₇	—	4	4	no	—
61	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
62	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
63	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
64	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
65	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
66	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
67	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
68	C₂³×C₄×C₆₇	C₂³	1	8	yes	—
69	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
70	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
71	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
72	#124×C₆₇	—	2	32	no	—
73	C₄×D₄×C₆₇	—	2	32	no	—
74	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
75	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
76	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
77	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
78	C₂×C₄⋊C₄×C₆₇	—	4	4	no	—
79	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
80	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
81	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
82	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
83	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
84	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
85	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
86	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
87	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
88	C₂×C₂²⋊C₄×C₆₇	—	4	4	no	—
89	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
90	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
91	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
92	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
93	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
94	C₂×C₄²×C₆₇	—	2	4	no	—
95	C₂×C₄²×C₆₇	—	2	4	no	—
96	C₂×C₄²×C₆₇	—	2	4	no	—
97	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
98	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
99	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
100	#95×C₆₇	C₂³	1	4	yes	—
101	#95×C₆₇	C₂³	1	4	yes	—
102	#95×C₆₇	C₂³	1	8	yes	—
103	#95×C₆₇	C₂³	1	8	yes	—
104	#95×C₆₇	C₂³	1	8	yes	—
105	#95×C₆₇	C₂³	1	8	yes	—
106	#95×C₆₇	C₂³	1	8	yes	—
107	C₂⁴×C₆₇	—	2	4	no	—
108	C₂⁴×C₆₇	—	2	4	no	—
109	C₂⁴×C₆₇	—	2	4	no	—
110	C₂⁴×C₆₇	—	2	4	no	—
111	C₂⁴×C₆₇	—	2	4	no	—
112	C₂⁴×C₆₇	—	2	4	no	—
113	C₂⁴×C₆₇	C₂×D₄	1	8	yes	—
114	C₂⁴×C₆₇	C₂⁴	1	4	yes	—
115	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
116	C₂×Q₈×C₆₇	—	2	8	no	—
117	C₂×Q₈×C₆₇	—	2	8	no	—
118	C₂×Q₈×C₆₇	—	2	8	no	—
119	C₂×Q₈×C₆₇	—	4	16	no	—
120	C₂×D₄×C₆₇	—	2	32	no	—
121	C₂×D₄×C₆₇	—	2	8	no	—
122	C₂×D₄×C₆₇	—	2	8	no	—
123	C₂×D₄×C₆₇	C₂×D₄	1	16	yes	—
124	C₂²×C₄×C₆₇	—	2	2	no	—
125	C₂²×C₄×C₆₇	—	2	2	no	—
126	C₂²×C₄×C₆₇	—	2	2	no	—
127	C₂²×C₄×C₆₇	—	2	2	no	—
128	C₂²×C₄×C₆₇	—	2	2	no	—
129	C₂²×C₄×C₆₇	—	2	2	no	—
130	C₂²×C₄×C₆₇	—	2	2	no	—
131	C₂²×C₄×C₆₇	—	2	4	no	—
132	C₂²×C₄×C₆₇	—	2	4	no	—
133	C₂²×C₄×C₆₇	—	2	16	no	—
134	C₂²×C₄×C₆₇	—	2	2	no	—
135	C₂²×C₄×C₆₇	—	2	4	no	—
136	C₂²×C₄×C₆₇	—	2	2	no	—
137	C₂²×C₄×C₆₇	—	2	2	no	—
138	C₂²×C₄×C₆₇	—	2	2	no	—
139	C₂²×C₄×C₆₇	—	2	4	no	—
140	C₂²×C₄×C₆₇	—	2	4	no	—
141	C₂²×C₄×C₆₇	—	2	4	no	—
142	C₂²×C₄×C₆₇	—	2	4	no	—
143	C₂²×C₄×C₆₇	—	2	2	no	—
144	C₂²×C₄×C₆₇	—	2	4	no	—
145	C₂²×C₄×C₆₇	—	2	2	no	—
146	C₂²×C₄×C₆₇	—	2	4	no	—
147	C₂²×C₄×C₆₇	—	2	2	no	—
148	C₂²×C₄×C₆₇	—	2	2	no	—
149	C₂²×C₄×C₆₇	—	2	2	no	—
150	C₂²×C₄×C₆₇	—	2	16	no	—
151	C₂²×C₄×C₆₇	—	4	8	no	—
152	C₂²×C₄×C₆₇	—	4	32	no	—
153	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
154	C₂²×C₄×C₆₇	#41	1	4	yes	—
155	C₂²×C₄×C₆₇	#41	1	4	yes	—
156	C₂²×C₄×C₆₇	#41	1	4	yes	—
157	C₂²×C₄×C₆₇	#41	1	4	yes	—
158	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
159	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
160	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
161	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
162	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
163	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
164	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
165	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
166	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
167	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
168	C₄⋊C₄×C₆₇	—	2	8	no	—
169	C₄⋊C₄×C₆₇	—	2	8	no	—
170	C₄⋊C₄×C₆₇	—	2	8	no	—
171	C₄⋊C₄×C₆₇	—	2	8	no	—
172	C₄⋊C₄×C₆₇	—	4	8	no	—
173	C₂²⋊C₄×C₆₇	—	2	16	no	—
174	C₂²⋊C₄×C₆₇	—	2	16	no	—
175	C₂²⋊C₄×C₆₇	—	2	16	no	—
176	C₂²⋊C₄×C₆₇	—	2	16	no	—
177	C₂²⋊C₄×C₆₇	—	4	8	no	—
178	C₄²×C₆₇	—	2	8	no	—
179	C₄²×C₆₇	—	2	8	no	—
180	C₄²×C₆₇	—	2	8	no	—
181	C₂³×C₆₇	—	2	16	no	—
182	C₂³×C₆₇	—	2	16	no	—
183	C₂³×C₆₇	—	2	16	no	—
184	C₂³×C₆₇	—	2	4	no	—
185	C₂³×C₆₇	—	2	4	no	—
186	C₂³×C₆₇	—	2	4	no	—
187	C₂³×C₆₇	—	2	4	no	—
188	C₂³×C₆₇	—	2	4	no	—
189	C₂³×C₆₇	—	2	4	no	—
190	C₂³×C₆₇	—	4	8	no	—
191	C₂³×C₆₇	—	4	8	no	—
192	C₂³×C₆₇	—	4	8	no	—
193	C₂³×C₆₇	—	4	8	no	—
194	C₂³×C₆₇	C₂²×D₄	1	8	yes	—
195	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
196	C₂³×C₆₇	C₂×#41	1	2	yes	—
197	C₂³×C₆₇	C₂×#41	1	2	yes	—
198	C₂³×C₆₇	C₂×#41	1	2	yes	—
199	C₂³×C₆₇	C₂×#41	1	2	yes	—
200	C₂³×C₆₇	H(5,𝔽₂)	1	2	yes	—
201	C₂³×C₆₇	2₋¹⁺⁴	1	2	yes	—
202	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
203	Q₈×C₆₇	—	4	16	no	—
204	D₄×C₆₇	—	2	32	no	—
205	C₂×C₄×C₆₇	—	2	8	no	—
206	C₂×C₄×C₆₇	—	2	8	no	—
207	C₂×C₄×C₆₇	—	2	2	no	—
208	C₂×C₄×C₆₇	—	2	2	no	—
209	C₂×C₄×C₆₇	—	2	2	no	—
210	C₂×C₄×C₆₇	—	2	2	no	—
211	C₂×C₄×C₆₇	—	2	2	no	—
212	C₂×C₄×C₆₇	—	2	2	no	—
213	C₂×C₄×C₆₇	—	2	2	no	—
214	C₂×C₄×C₆₇	—	2	4	no	—
215	C₂×C₄×C₆₇	—	2	4	no	—
216	C₂×C₄×C₆₇	—	2	2	no	—
217	C₂×C₄×C₆₇	—	2	4	no	—
218	C₂×C₄×C₆₇	—	2	2	no	—
219	C₂×C₄×C₆₇	—	2	2	no	—
220	C₂×C₄×C₆₇	—	2	2	no	—
221	C₂×C₄×C₆₇	—	2	4	no	—
222	C₂×C₄×C₆₇	—	2	4	no	—
223	C₂×C₄×C₆₇	—	2	4	no	—
224	C₂×C₄×C₆₇	—	2	4	no	—
225	C₂×C₄×C₆₇	—	2	2	no	—
226	C₂×C₄×C₆₇	—	2	4	no	—
227	C₂×C₄×C₆₇	—	2	2	no	—
228	C₂×C₄×C₆₇	—	2	4	no	—
229	C₂×C₄×C₆₇	—	2	2	no	—
230	C₂×C₄×C₆₇	—	2	2	no	—
231	C₂×C₄×C₆₇	—	2	2	no	—
232	C₂×C₄×C₆₇	—	4	32	no	—
233	C₂×C₄×C₆₇	—	4	8	no	—
234	C₂×C₄×C₆₇	—	4	4	no	—
235	C₂×C₄×C₆₇	—	4	4	no	—
236	C₂×C₄×C₆₇	—	4	8	no	—
237	C₂×C₄×C₆₇	—	4	4	no	—
238	C₂×C₄×C₆₇	—	4	4	no	—
239	C₂×C₄×C₆₇	—	4	8	no	—
240	C₂×C₄×C₆₇	#123	1	16	yes	—
241	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
242	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
243	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
244	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
245	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
246	C₂²×C₆₇	—	2	8	no	—
247	C₂²×C₆₇	—	2	8	no	—
248	C₂²×C₆₇	—	2	8	no	—
249	C₂²×C₆₇	—	4	4	no	—
250	C₂²×C₆₇	—	4	4	no	—
251	C₂²×C₆₇	—	4	32	no	—
252	C₂²×C₆₇	—	4	4	no	—
253	C₂²×C₆₇	—	4	4	no	—
254	C₂²×C₆₇	C₂×#123	1	4	yes	—
255	C₂²×C₆₇	#529	1	4	yes	—
256	C₂²×C₆₇	#542	1	4	yes	—
257	C₂²×C₆₇	#546	1	4	yes	—
258	C₂²×C₆₇	#547	1	4	yes	—
259	C₂²×C₆₇	#556	1	4	yes	—
260	C₂²×C₆₇	#559	1	4	yes	—
261	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
262	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
263	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
264	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
265	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
266	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
267	C₂²×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
268	C₄×C₆₇	—	2	4	no	—
269	C₄×C₆₇	—	2	4	no	—
270	C₄×C₆₇	—	4	4	no	—
271	C₄×C₆₇	—	4	2	no	—
272	C₄×C₆₇	—	4	2	no	—
273	C₄×C₆₇	—	4	4	no	—
274	C₄×C₆₇	—	4	2	no	—
275	C₄×C₆₇	—	4	2	no	—
276	C₄×C₆₇	—	4	4	no	—
277	C₄×C₆₇	C₂×#123	1	8	yes	—
278	C₂×#2447	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
279	C₂×C₆₇	—	4	8	no	—
280	C₂×C₆₇	#2447	1	8	yes	direct product factor
281	C₂×C₆₇	C₂²×#123	1	1	yes	characteristic
282	C₂×C₆₇	C₂×#529	1	1	yes	characteristic
283	C₂×C₆₇	C₂×#542	1	1	yes	characteristic
284	C₂×C₆₇	C₂×#546	1	1	yes	characteristic
285	C₂×C₆₇	C₂×#547	1	1	yes	characteristic
286	C₂×C₆₇	C₂×#556	1	1	yes	characteristic
287	C₂×C₆₇	C₂×#559	1	1	yes	characteristic
288	C₂²×#124	C₂×C₆₇	1	1	yes	characteristic
289	C₂²×C₄×D₄	C₂×C₆₇	1	1	yes	characteristic
290	#2447	C₂×C₆₇	1	16	yes	direct product factor
291	C₂×#397	C₂×C₆₇	1	2	yes	complemented
292	C₂×#397	C₂×C₆₇	1	2	yes	complemented
293	C₂×#391	C₂×C₆₇	1	1	yes	characteristic, complemented
294	C₂×#391	C₂×C₆₇	1	1	yes	characteristic, complemented
295	C₂×#390	C₂×C₆₇	1	1	yes	characteristic
296	C₂×#389	C₂×C₆₇	1	1	yes	characteristic, complemented
297	C₂×#385	C₂×C₆₇	1	2	yes	complemented
298	C₂×#381	C₂×C₆₇	1	2	yes	complemented
299	C₂×C₄×C₄⋊C₄	C₂×C₆₇	1	1	yes	characteristic, complemented
300	C₆₇	C₂×#2447	1	1	yes	characteristic, direct product factor, Sylow
301	C₂³×Q₈	C₂²×C₆₇	1	1	yes	characteristic
302	C₂³×D₄	C₂²×C₆₇	1	1	yes	characteristic
303	C₂⁴×C₄	C₂²×C₆₇	1	2	yes	—
304	C₂×#124	—	2	8	no	—
305	C₂×#124	C₂²×C₆₇	1	8	yes	—
306	C₂×C₄×D₄	—	2	8	no	—
307	C₂×C₄×D₄	C₂²×C₆₇	1	8	yes	—
308	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
309	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
310	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
311	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
312	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
313	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
314	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
315	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
316	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
317	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
318	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
319	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
320	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
321	#397	C₂²×C₆₇	1	16	yes	complemented
322	#397	C₂²×C₆₇	1	16	yes	complemented
323	#391	C₂²×C₆₇	1	8	yes	complemented
324	#391	C₂²×C₆₇	1	8	yes	complemented
325	#390	C₂²×C₆₇	1	8	yes	—
326	#389	C₂²×C₆₇	1	8	yes	complemented
327	#385	C₂²×C₆₇	1	16	yes	complemented
328	#381	C₂²×C₆₇	1	16	yes	complemented
329	C₄×C₄⋊C₄	C₂²×C₆₇	1	8	yes	complemented
330	C₂×#95	C₂²×C₆₇	1	1	yes	characteristic
331	C₂×#95	C₂²×C₆₇	1	2	yes	—
332	C₂×#95	C₂²×C₆₇	1	2	yes	—
333	C₂×#95	C₂²×C₆₇	1	2	yes	—
334	C₂×#95	C₂²×C₆₇	1	2	yes	—
335	C₂×#95	C₂²×C₆₇	1	1	yes	characteristic
336	C₂×#95	C₂²×C₆₇	1	2	yes	—
337	C₂⁵	C₂³×C₆₇	1	2	yes	—
338	C₂²×Q₈	—	2	4	no	—
339	C₂²×Q₈	—	2	4	no	—
340	C₂²×Q₈	—	2	4	no	—
341	C₂²×Q₈	C₂³×C₆₇	1	4	yes	—
342	C₂²×D₄	—	2	4	no	—
343	C₂²×D₄	—	2	4	no	—
344	C₂²×D₄	D₄×C₆₇	1	8	yes	—
345	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
346	C₂³×C₄	—	2	8	no	—
347	C₂³×C₄	—	2	8	no	—
348	C₂³×C₄	—	4	4	no	—
349	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
350	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
351	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
352	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
353	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
354	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
355	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
356	C₂³×C₄	C₂³×C₆₇	1	8	yes	—
357	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
358	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
359	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
360	#124	—	2	32	no	—
361	C₄×D₄	—	2	32	no	—
362	C₂×C₄⋊C₄	—	2	4	no	—
363	C₂×C₄⋊C₄	—	2	4	no	—
364	C₂×C₄⋊C₄	—	2	4	no	—
365	C₂×C₄⋊C₄	—	2	4	no	—
366	C₂×C₄⋊C₄	—	4	4	no	—
367	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
368	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
369	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
370	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
371	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
372	C₂×C₂²⋊C₄	—	2	8	no	—
373	C₂×C₂²⋊C₄	—	2	8	no	—
374	C₂×C₂²⋊C₄	—	2	8	no	—
375	C₂×C₂²⋊C₄	—	2	8	no	—
376	C₂×C₂²⋊C₄	—	4	4	no	—
377	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
378	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
379	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
380	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
381	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
382	C₂×C₄²	—	2	4	no	—
383	C₂×C₄²	—	2	4	no	—
384	C₂×C₄²	—	2	4	no	—
385	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
386	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
387	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
388	#95	C₂³×C₆₇	1	4	yes	—
389	#95	C₂³×C₆₇	1	4	yes	—
390	#95	C₂³×C₆₇	1	8	yes	—
391	#95	C₂³×C₆₇	1	8	yes	—
392	#95	C₂³×C₆₇	1	8	yes	—
393	#95	C₂³×C₆₇	1	8	yes	—
394	#95	C₂³×C₆₇	1	8	yes	—
395	C₂⁴	—	2	4	no	—
396	C₂⁴	—	2	4	no	—
397	C₂⁴	—	2	4	no	—
398	C₂⁴	—	2	4	no	—
399	C₂⁴	—	2	4	no	—
400	C₂⁴	—	2	4	no	—
401	C₂⁴	C₂×D₄×C₆₇	1	8	yes	—
402	C₂⁴	C₂⁴×C₆₇	1	4	yes	—
403	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
404	C₂×Q₈	—	2	8	no	—
405	C₂×Q₈	—	2	8	no	—
406	C₂×Q₈	—	2	8	no	—
407	C₂×Q₈	—	4	16	no	—
408	C₂×D₄	—	2	32	no	—
409	C₂×D₄	—	2	8	no	—
410	C₂×D₄	—	2	8	no	—
411	C₂×D₄	C₂×D₄×C₆₇	1	16	yes	—
412	C₂²×C₄	—	2	2	no	—
413	C₂²×C₄	—	2	2	no	—
414	C₂²×C₄	—	2	2	no	—
415	C₂²×C₄	—	2	2	no	—
416	C₂²×C₄	—	2	2	no	—
417	C₂²×C₄	—	2	2	no	—
418	C₂²×C₄	—	2	2	no	—
419	C₂²×C₄	—	2	4	no	—
420	C₂²×C₄	—	2	4	no	—
421	C₂²×C₄	—	2	16	no	—
422	C₂²×C₄	—	2	2	no	—
423	C₂²×C₄	—	2	4	no	—
424	C₂²×C₄	—	2	2	no	—
425	C₂²×C₄	—	2	2	no	—
426	C₂²×C₄	—	2	2	no	—
427	C₂²×C₄	—	2	4	no	—
428	C₂²×C₄	—	2	4	no	—
429	C₂²×C₄	—	2	4	no	—
430	C₂²×C₄	—	2	4	no	—
431	C₂²×C₄	—	2	2	no	—
432	C₂²×C₄	—	2	4	no	—
433	C₂²×C₄	—	2	2	no	—
434	C₂²×C₄	—	2	4	no	—
435	C₂²×C₄	—	2	2	no	—
436	C₂²×C₄	—	2	2	no	—
437	C₂²×C₄	—	2	2	no	—
438	C₂²×C₄	—	2	16	no	—
439	C₂²×C₄	—	4	8	no	—
440	C₂²×C₄	—	4	32	no	—
441	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
442	C₂²×C₄	#41×C₆₇	1	4	yes	—
443	C₂²×C₄	#41×C₆₇	1	4	yes	—
444	C₂²×C₄	#41×C₆₇	1	4	yes	—
445	C₂²×C₄	#41×C₆₇	1	4	yes	—
446	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
447	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
448	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
449	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
450	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
451	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
452	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
453	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
454	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
455	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
456	C₄⋊C₄	—	2	8	no	—
457	C₄⋊C₄	—	2	8	no	—
458	C₄⋊C₄	—	2	8	no	—
459	C₄⋊C₄	—	2	8	no	—
460	C₄⋊C₄	—	4	8	no	—
461	C₂²⋊C₄	—	2	16	no	—
462	C₂²⋊C₄	—	2	16	no	—
463	C₂²⋊C₄	—	2	16	no	—
464	C₂²⋊C₄	—	2	16	no	—
465	C₂²⋊C₄	—	4	8	no	—
466	C₄²	—	2	8	no	—
467	C₄²	—	2	8	no	—
468	C₄²	—	2	8	no	—
469	C₂³	—	2	16	no	—
470	C₂³	—	2	16	no	—
471	C₂³	—	2	16	no	—
472	C₂³	—	2	4	no	—
473	C₂³	—	2	4	no	—
474	C₂³	—	2	4	no	—
475	C₂³	—	2	4	no	—
476	C₂³	—	2	4	no	—
477	C₂³	—	2	4	no	—
478	C₂³	—	4	8	no	—
479	C₂³	—	4	8	no	—
480	C₂³	—	4	8	no	—
481	C₂³	—	4	8	no	—
482	C₂³	C₂²×D₄×C₆₇	1	8	yes	—
483	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
484	C₂³	C₂×#41×C₆₇	1	2	yes	—
485	C₂³	C₂×#41×C₆₇	1	2	yes	—
486	C₂³	C₂×#41×C₆₇	1	2	yes	—
487	C₂³	C₂×#41×C₆₇	1	2	yes	—
488	C₂³	H(5,𝔽₂)×C₆₇	1	2	yes	—
489	C₂³	2₋¹⁺⁴×C₆₇	1	2	yes	—
490	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
491	Q₈	—	4	16	no	—
492	D₄	—	2	32	no	—
493	C₂×C₄	—	2	8	no	—
494	C₂×C₄	—	2	8	no	—
495	C₂×C₄	—	2	2	no	—
496	C₂×C₄	—	2	2	no	—
497	C₂×C₄	—	2	2	no	—
498	C₂×C₄	—	2	2	no	—
499	C₂×C₄	—	2	2	no	—
500	C₂×C₄	—	2	2	no	—
501	C₂×C₄	—	2	2	no	—
502	C₂×C₄	—	2	4	no	—
503	C₂×C₄	—	2	4	no	—
504	C₂×C₄	—	2	2	no	—
505	C₂×C₄	—	2	4	no	—
506	C₂×C₄	—	2	2	no	—
507	C₂×C₄	—	2	2	no	—
508	C₂×C₄	—	2	2	no	—
509	C₂×C₄	—	2	4	no	—
510	C₂×C₄	—	2	4	no	—
511	C₂×C₄	—	2	4	no	—
512	C₂×C₄	—	2	4	no	—
513	C₂×C₄	—	2	2	no	—
514	C₂×C₄	—	2	4	no	—
515	C₂×C₄	—	2	2	no	—
516	C₂×C₄	—	2	4	no	—
517	C₂×C₄	—	2	2	no	—
518	C₂×C₄	—	2	2	no	—
519	C₂×C₄	—	2	2	no	—
520	C₂×C₄	—	4	32	no	—
521	C₂×C₄	—	4	8	no	—
522	C₂×C₄	—	4	4	no	—
523	C₂×C₄	—	4	8	no	—
524	C₂×C₄	—	4	4	no	—
525	C₂×C₄	—	4	4	no	—
526	C₂×C₄	—	4	4	no	—
527	C₂×C₄	—	4	8	no	—
528	C₂×C₄	#123×C₆₇	1	16	yes	—
529	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
530	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
531	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
532	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
533	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
534	C₂²	—	2	8	no	—
535	C₂²	—	2	8	no	—
536	C₂²	—	2	8	no	—
537	C₂²	—	4	4	no	—
538	C₂²	—	4	4	no	—
539	C₂²	—	4	32	no	—
540	C₂²	—	4	4	no	—
541	C₂²	—	4	4	no	—
542	C₂²	C₂×#123×C₆₇	1	4	yes	—
543	C₂²	#529×C₆₇	1	4	yes	—
544	C₂²	#542×C₆₇	1	4	yes	—
545	C₂²	#546×C₆₇	1	4	yes	—
546	C₂²	#547×C₆₇	1	4	yes	—
547	C₂²	#556×C₆₇	1	4	yes	—
548	C₂²	#559×C₆₇	1	4	yes	—
549	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
550	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
551	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
552	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
553	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
554	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
555	C₂²	C₂×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
556	C₄	—	2	4	no	—
557	C₄	—	2	4	no	—
558	C₄	—	4	4	no	—
559	C₄	—	4	2	no	—
560	C₄	—	4	4	no	—
561	C₄	—	4	2	no	—
562	C₄	—	4	2	no	—
563	C₄	—	4	2	no	—
564	C₄	—	4	4	no	—
565	C₄	C₂×#123×C₆₇	1	8	yes	—
566	C₂	—	4	8	no	—
567	C₂	C₆₇×#2447	1	8	yes	direct product factor
568	C₂	C₂²×#123×C₆₇	1	1	yes	characteristic
569	C₂	C₂×#529×C₆₇	1	1	yes	characteristic
570	C₂	C₂×#542×C₆₇	1	1	yes	characteristic
571	C₂	C₂×#546×C₆₇	1	1	yes	characteristic
572	C₂	C₂×#547×C₆₇	1	1	yes	characteristic
573	C₂	C₂×#556×C₆₇	1	1	yes	characteristic
574	C₂	C₂×#559×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₆₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₆₇
prime radical	C₂³×D₄×C₆₇	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₆₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
3. C₆₇×#2447	C₂	direct (N × H)
4. C₂×#397×C₆₇	C₂	semidirect
5. C₂×#397×C₆₇	C₂	semidirect
6. C₂×#391×C₆₇	C₂	semidirect
7. C₂×#391×C₆₇	C₂	semidirect
9. C₂×#389×C₆₇	C₂	semidirect
10. C₂×#385×C₆₇	C₂	semidirect
11. C₂×#381×C₆₇	C₂	semidirect
12. C₂×C₄×C₄⋊C₄×C₆₇	C₂	semidirect
33. #397×C₆₇	C₂²	semidirect
34. #397×C₆₇	C₂²	semidirect
35. #391×C₆₇	C₂²	semidirect
36. #391×C₆₇	C₂²	semidirect
38. #389×C₆₇	C₂²	semidirect
39. #385×C₆₇	C₂²	semidirect
40. #381×C₆₇	C₂²	semidirect
41. C₄×C₄⋊C₄×C₆₇	C₂²	semidirect
278. C₂×#2447	C₆₇	direct (N × H)
280. C₂×C₆₇	#2447	direct (N × H)
290. #2447	C₂×C₆₇	direct (N × H)
291. C₂×#397	C₂×C₆₇	semidirect
292. C₂×#397	C₂×C₆₇	semidirect
293. C₂×#391	C₂×C₆₇	semidirect
294. C₂×#391	C₂×C₆₇	semidirect
296. C₂×#389	C₂×C₆₇	semidirect
297. C₂×#385	C₂×C₆₇	semidirect
298. C₂×#381	C₂×C₆₇	semidirect
299. C₂×C₄×C₄⋊C₄	C₂×C₆₇	semidirect
300. C₆₇	C₂×#2447	direct (N × H)
321. #397	C₂²×C₆₇	semidirect
322. #397	C₂²×C₆₇	semidirect
323. #391	C₂²×C₆₇	semidirect
324. #391	C₂²×C₆₇	semidirect
326. #389	C₂²×C₆₇	semidirect
327. #385	C₂²×C₆₇	semidirect
328. #381	C₂²×C₆₇	semidirect
329. C₄×C₄⋊C₄	C₂²×C₆₇	semidirect
567. C₂	C₆₇×#2447	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×#124×C₆₇	2
C₂²×C₄×D₄×C₆₇	2
C₆₇×#2447	2
C₂×#397×C₆₇	2
C₂×#391×C₆₇	2
C₂×#390×C₆₇	2
C₂×#389×C₆₇	2
C₂×#385×C₆₇	2
C₂×#381×C₆₇	2
C₂×C₄×C₄⋊C₄×C₆₇	2
C₂×#2447	67

Subgroups — C2×C67×#2447 #30713949

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#2447 #30713949