Subgroups — C₂×C₆₇×#2337 #30713331

Order 17152.

3046 conjugacy classes of subgroups.

The subgroup lattice has 476 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	C₆₇×#2337	C₂	1	16	yes	maximal, direct product factor
3	C₂×#391×C₆₇	C₂	1	2	yes	maximal, complemented
4	C₂×#388×C₆₇	C₂	1	2	yes	maximal, complemented
5	C₂×#388×C₆₇	C₂	1	2	yes	maximal, complemented
6	C₂×#386×C₆₇	C₂	1	2	yes	maximal
7	C₂×#385×C₆₇	C₂	1	2	yes	maximal, complemented
8	C₂×#383×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₄×C₄⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₄×C₂²⋊C₄×C₆₇	C₂	1	2	yes	maximal, complemented
11	C₂³×Q₈×C₆₇	C₂²	1	1	yes	characteristic
12	C₂³×D₄×C₆₇	C₂²	1	1	yes	characteristic
13	C₂⁴×C₄×C₆₇	C₂²	1	2	yes	—
14	C₂×#124×C₆₇	C₄	1	16	yes	complemented
15	C₂×#124×C₆₇	C₂²	1	8	yes	—
16	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
17	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
18	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
19	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
20	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
21	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
22	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
23	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
24	C₂²×C₄²×C₆₇	C₂²	1	2	yes	—
25	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
26	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
27	#391×C₆₇	C₂²	1	16	yes	complemented
28	#388×C₆₇	C₂²	1	16	yes	complemented
29	#388×C₆₇	C₂²	1	16	yes	complemented
30	#386×C₆₇	C₂²	1	16	yes	—
31	#385×C₆₇	C₂²	1	16	yes	complemented
32	#383×C₆₇	C₂²	1	8	yes	complemented
33	C₄×C₄⋊C₄×C₆₇	C₂²	1	8	yes	complemented
34	C₄×C₂²⋊C₄×C₆₇	C₂²	1	16	yes	complemented
35	C₂×#95×C₆₇	C₂²	1	2	yes	—
36	C₂×#95×C₆₇	C₂²	1	2	yes	—
37	C₂×#95×C₆₇	C₂²	1	2	yes	—
38	C₂×#95×C₆₇	C₂²	1	1	yes	characteristic
39	C₂×#95×C₆₇	C₂²	1	2	yes	—
40	C₂×#95×C₆₇	C₂²	1	1	yes	characteristic
41	C₂×#95×C₆₇	C₂²	1	2	yes	—
42	C₂⁵×C₆₇	C₂³	1	2	yes	—
43	C₂²×Q₈×C₆₇	—	4	4	no	—
44	C₂²×Q₈×C₆₇	C₂×C₄	1	8	yes	complemented
45	C₂²×Q₈×C₆₇	C₂³	1	4	yes	—
46	C₂²×D₄×C₆₇	—	4	4	no	—
47	C₂²×D₄×C₆₇	C₂×C₄	1	8	yes	—
48	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
49	C₂³×C₄×C₆₇	—	2	8	no	—
50	C₂³×C₄×C₆₇	—	4	4	no	—
51	C₂³×C₄×C₆₇	—	4	4	no	—
52	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
53	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
54	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
55	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
56	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
57	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
58	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
59	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
60	C₂³×C₄×C₆₇	C₂³	1	8	yes	—
61	#124×C₆₇	C₂×C₄	1	64	yes	complemented
62	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
63	C₂×C₄⋊C₄×C₆₇	—	4	4	no	—
64	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
65	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
66	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
67	C₂×C₂²⋊C₄×C₆₇	—	4	4	no	—
68	C₂×C₂²⋊C₄×C₆₇	—	4	4	no	—
69	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	16	yes	complemented
70	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	16	yes	complemented
71	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
72	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
73	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
74	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
75	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
76	C₂×C₄²×C₆₇	—	4	2	no	—
77	C₂×C₄²×C₆₇	—	4	2	no	—
78	C₂×C₄²×C₆₇	—	4	4	no	—
79	C₂×C₄²×C₆₇	C₂×C₄	1	8	yes	complemented
80	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
81	C₂×C₄²×C₆₇	C₂³	1	8	yes	—
82	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
83	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
84	#95×C₆₇	C₂³	1	8	yes	—
85	#95×C₆₇	C₂³	1	8	yes	—
86	#95×C₆₇	C₂³	1	8	yes	—
87	#95×C₆₇	C₂³	1	4	yes	—
88	#95×C₆₇	C₂³	1	8	yes	—
89	#95×C₆₇	C₂³	1	4	yes	—
90	#95×C₆₇	C₂³	1	8	yes	—
91	C₂⁴×C₆₇	—	2	4	no	—
92	C₂⁴×C₆₇	—	2	4	no	—
93	C₂⁴×C₆₇	—	2	4	no	—
94	C₂⁴×C₆₇	—	2	4	no	—
95	C₂⁴×C₆₇	—	2	4	no	—
96	C₂⁴×C₆₇	—	2	4	no	—
97	C₂⁴×C₆₇	C₂²×C₄	1	8	yes	—
98	C₂⁴×C₆₇	C₂⁴	1	4	yes	—
99	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
100	C₂×Q₈×C₆₇	—	4	16	no	—
101	C₂×Q₈×C₆₇	—	4	8	no	—
102	C₂×Q₈×C₆₇	C₂²×C₄	1	16	yes	complemented
103	C₂×D₄×C₆₇	—	4	16	no	—
104	C₂×D₄×C₆₇	—	4	8	no	—
105	C₂×D₄×C₆₇	C₂²×C₄	1	16	yes	—
106	C₂²×C₄×C₆₇	—	2	4	no	—
107	C₂²×C₄×C₆₇	—	2	2	no	—
108	C₂²×C₄×C₆₇	—	2	4	no	—
109	C₂²×C₄×C₆₇	—	2	4	no	—
110	C₂²×C₄×C₆₇	—	2	4	no	—
111	C₂²×C₄×C₆₇	—	2	4	no	—
112	C₂²×C₄×C₆₇	—	2	4	no	—
113	C₂²×C₄×C₆₇	—	2	2	no	—
114	C₂²×C₄×C₆₇	—	2	4	no	—
115	C₂²×C₄×C₆₇	—	2	4	no	—
116	C₂²×C₄×C₆₇	—	2	4	no	—
117	C₂²×C₄×C₆₇	—	2	4	no	—
118	C₂²×C₄×C₆₇	—	2	4	no	—
119	C₂²×C₄×C₆₇	—	2	4	no	—
120	C₂²×C₄×C₆₇	—	2	4	no	—
121	C₂²×C₄×C₆₇	—	2	4	no	—
122	C₂²×C₄×C₆₇	—	2	4	no	—
123	C₂²×C₄×C₆₇	—	2	16	no	—
124	C₂²×C₄×C₆₇	—	4	8	no	—
125	C₂²×C₄×C₆₇	—	4	8	no	—
126	C₂²×C₄×C₆₇	—	4	32	no	—
127	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
128	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
129	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
130	C₂²×C₄×C₆₇	C₂²×C₄	1	8	yes	—
131	C₂²×C₄×C₆₇	#41	1	8	yes	—
132	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
133	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
134	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
135	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
136	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
137	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
138	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
139	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
140	C₄⋊C₄×C₆₇	—	2	16	no	—
141	C₄⋊C₄×C₆₇	—	4	8	no	—
142	C₂²⋊C₄×C₆₇	—	2	16	no	—
143	C₂²⋊C₄×C₆₇	—	4	8	no	—
144	C₂²⋊C₄×C₆₇	—	4	8	no	—
145	C₂²⋊C₄×C₆₇	C₂²×C₄	1	32	yes	complemented
146	C₂²⋊C₄×C₆₇	C₂²×C₄	1	32	yes	complemented
147	C₄²×C₆₇	—	4	4	no	—
148	C₄²×C₆₇	—	4	4	no	—
149	C₄²×C₆₇	—	4	8	no	—
150	C₄²×C₆₇	C₂²×C₄	1	16	yes	complemented
151	C₂³×C₆₇	—	2	16	no	—
152	C₂³×C₆₇	—	2	16	no	—
153	C₂³×C₆₇	—	2	16	no	—
154	C₂³×C₆₇	—	2	4	no	—
155	C₂³×C₆₇	—	2	4	no	—
156	C₂³×C₆₇	—	2	4	no	—
157	C₂³×C₆₇	—	2	4	no	—
158	C₂³×C₆₇	—	2	4	no	—
159	C₂³×C₆₇	—	2	4	no	—
160	C₂³×C₆₇	—	4	8	no	—
161	C₂³×C₆₇	—	4	8	no	—
162	C₂³×C₆₇	—	4	8	no	—
163	C₂³×C₆₇	—	4	8	no	—
164	C₂³×C₆₇	C₂³×C₄	1	8	yes	—
165	C₂³×C₆₇	C₂³×C₄	1	2	yes	—
166	C₂³×C₆₇	C₂×#41	1	4	yes	—
167	C₂³×C₆₇	H(5,𝔽₂)	1	4	yes	—
168	C₂³×C₆₇	H(5,𝔽₂)	1	2	yes	—
169	C₂³×C₆₇	2₋¹⁺⁴	1	2	yes	—
170	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
171	Q₈×C₆₇	—	4	16	no	—
172	D₄×C₆₇	—	4	16	no	—
173	C₂×C₄×C₆₇	—	2	8	no	—
174	C₂×C₄×C₆₇	—	2	8	no	—
175	C₂×C₄×C₆₇	—	2	16	no	—
176	C₂×C₄×C₆₇	—	2	8	no	—
177	C₂×C₄×C₆₇	—	2	16	no	—
178	C₂×C₄×C₆₇	—	2	4	no	—
179	C₂×C₄×C₆₇	—	2	2	no	—
180	C₂×C₄×C₆₇	—	2	4	no	—
181	C₂×C₄×C₆₇	—	2	4	no	—
182	C₂×C₄×C₆₇	—	2	4	no	—
183	C₂×C₄×C₆₇	—	2	4	no	—
184	C₂×C₄×C₆₇	—	2	4	no	—
185	C₂×C₄×C₆₇	—	2	2	no	—
186	C₂×C₄×C₆₇	—	2	4	no	—
187	C₂×C₄×C₆₇	—	2	4	no	—
188	C₂×C₄×C₆₇	—	2	4	no	—
189	C₂×C₄×C₆₇	—	2	4	no	—
190	C₂×C₄×C₆₇	—	2	4	no	—
191	C₂×C₄×C₆₇	—	2	4	no	—
192	C₂×C₄×C₆₇	—	2	4	no	—
193	C₂×C₄×C₆₇	—	2	4	no	—
194	C₂×C₄×C₆₇	—	2	4	no	—
195	C₂×C₄×C₆₇	—	4	32	no	—
196	C₂×C₄×C₆₇	—	4	8	no	—
197	C₂×C₄×C₆₇	—	4	8	no	—
198	C₂×C₄×C₆₇	—	4	8	no	—
199	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
200	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
201	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
202	C₂×C₄×C₆₇	C₂³×C₄	1	8	yes	—
203	C₂×C₄×C₆₇	C₂×#41	1	8	yes	—
204	C₂²×C₆₇	—	2	8	no	—
205	C₂²×C₆₇	—	2	8	no	—
206	C₂²×C₆₇	—	2	8	no	—
207	C₂²×C₆₇	—	4	32	no	—
208	C₂²×C₆₇	—	4	4	no	—
209	C₂²×C₆₇	—	4	4	no	—
210	C₂²×C₆₇	—	4	4	no	—
211	C₂²×C₆₇	—	4	4	no	—
212	C₂²×C₆₇	C₄×#41	1	4	yes	—
213	C₂²×C₆₇	#518	1	4	yes	—
214	C₂²×C₆₇	#520	1	4	yes	—
215	C₂²×C₆₇	#538	1	8	yes	—
216	C₂²×C₆₇	#542	1	8	yes	—
217	C₂²×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
218	C₂²×C₆₇	C₂²×#41	1	2	yes	—
219	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	2	yes	—
220	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
221	C₂²×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
222	C₄×C₆₇	—	2	4	no	—
223	C₄×C₆₇	—	2	4	no	—
224	C₄×C₆₇	—	2	8	no	—
225	C₄×C₆₇	—	2	4	no	—
226	C₄×C₆₇	—	2	8	no	—
227	C₄×C₆₇	—	4	4	no	—
228	C₄×C₆₇	—	4	4	no	—
229	C₄×C₆₇	—	4	4	no	—
230	C₂×#2337	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
231	C₂×C₆₇	—	4	8	no	—
232	C₂×C₆₇	#2337	1	8	yes	direct product factor
233	C₂×C₆₇	C₂×C₄×#41	1	1	yes	characteristic
234	C₂×C₆₇	C₂×#518	1	1	yes	characteristic
235	C₂×C₆₇	C₂×#520	1	1	yes	characteristic
236	C₂×C₆₇	C₂×#538	1	2	yes	—
237	C₂×C₆₇	C₂×#542	1	2	yes	—
238	C₂²×#124	C₂×C₆₇	1	1	yes	characteristic
239	#2337	C₂×C₆₇	1	16	yes	direct product factor
240	C₂×#391	C₂×C₆₇	1	2	yes	complemented
241	C₂×#388	C₂×C₆₇	1	2	yes	complemented
242	C₂×#388	C₂×C₆₇	1	2	yes	complemented
243	C₂×#386	C₂×C₆₇	1	2	yes	—
244	C₂×#385	C₂×C₆₇	1	2	yes	complemented
245	C₂×#383	C₂×C₆₇	1	1	yes	characteristic, complemented
246	C₂×C₄×C₄⋊C₄	C₂×C₆₇	1	1	yes	characteristic, complemented
247	C₂×C₄×C₂²⋊C₄	C₂×C₆₇	1	2	yes	complemented
248	C₆₇	C₂×#2337	1	1	yes	characteristic, direct product factor, Sylow
249	C₂³×Q₈	C₂²×C₆₇	1	1	yes	characteristic
250	C₂³×D₄	C₂²×C₆₇	1	1	yes	characteristic
251	C₂⁴×C₄	C₂²×C₆₇	1	2	yes	—
252	C₂×#124	C₄×C₆₇	1	16	yes	complemented
253	C₂×#124	C₂²×C₆₇	1	8	yes	—
254	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
255	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
256	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
257	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
258	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
259	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
260	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
261	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
262	C₂²×C₄²	C₂²×C₆₇	1	2	yes	—
263	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
264	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
265	#391	C₂²×C₆₇	1	16	yes	complemented
266	#388	C₂²×C₆₇	1	16	yes	complemented
267	#388	C₂²×C₆₇	1	16	yes	complemented
268	#386	C₂²×C₆₇	1	16	yes	—
269	#385	C₂²×C₆₇	1	16	yes	complemented
270	#383	C₂²×C₆₇	1	8	yes	complemented
271	C₄×C₄⋊C₄	C₂²×C₆₇	1	8	yes	complemented
272	C₄×C₂²⋊C₄	C₂²×C₆₇	1	16	yes	complemented
273	C₂×#95	C₂²×C₆₇	1	2	yes	—
274	C₂×#95	C₂²×C₆₇	1	2	yes	—
275	C₂×#95	C₂²×C₆₇	1	2	yes	—
276	C₂×#95	C₂²×C₆₇	1	1	yes	characteristic
277	C₂×#95	C₂²×C₆₇	1	2	yes	—
278	C₂×#95	C₂²×C₆₇	1	1	yes	characteristic
279	C₂×#95	C₂²×C₆₇	1	2	yes	—
280	C₂⁵	C₂³×C₆₇	1	2	yes	—
281	C₂²×Q₈	—	4	4	no	—
282	C₂²×Q₈	C₂×C₄×C₆₇	1	8	yes	complemented
283	C₂²×Q₈	C₂³×C₆₇	1	4	yes	—
284	C₂²×D₄	—	4	4	no	—
285	C₂²×D₄	C₂×C₄×C₆₇	1	8	yes	—
286	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
287	C₂³×C₄	—	2	8	no	—
288	C₂³×C₄	—	4	4	no	—
289	C₂³×C₄	—	4	4	no	—
290	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
291	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
292	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
293	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
294	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
295	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
296	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
297	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
298	C₂³×C₄	C₂³×C₆₇	1	8	yes	—
299	#124	C₂×C₄×C₆₇	1	64	yes	complemented
300	C₂×C₄⋊C₄	—	2	8	no	—
301	C₂×C₄⋊C₄	—	4	4	no	—
302	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
303	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
304	C₂×C₂²⋊C₄	—	2	8	no	—
305	C₂×C₂²⋊C₄	—	4	4	no	—
306	C₂×C₂²⋊C₄	—	4	4	no	—
307	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	16	yes	complemented
308	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	16	yes	complemented
309	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
310	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
311	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
312	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
313	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
314	C₂×C₄²	—	4	2	no	—
315	C₂×C₄²	—	4	2	no	—
316	C₂×C₄²	—	4	4	no	—
317	C₂×C₄²	C₂×C₄×C₆₇	1	8	yes	complemented
318	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
319	C₂×C₄²	C₂³×C₆₇	1	8	yes	—
320	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
321	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
322	#95	C₂³×C₆₇	1	8	yes	—
323	#95	C₂³×C₆₇	1	8	yes	—
324	#95	C₂³×C₆₇	1	8	yes	—
325	#95	C₂³×C₆₇	1	4	yes	—
326	#95	C₂³×C₆₇	1	8	yes	—
327	#95	C₂³×C₆₇	1	4	yes	—
328	#95	C₂³×C₆₇	1	8	yes	—
329	C₂⁴	—	2	4	no	—
330	C₂⁴	—	2	4	no	—
331	C₂⁴	—	2	4	no	—
332	C₂⁴	—	2	4	no	—
333	C₂⁴	—	2	4	no	—
334	C₂⁴	—	2	4	no	—
335	C₂⁴	C₂²×C₄×C₆₇	1	8	yes	—
336	C₂⁴	C₂⁴×C₆₇	1	4	yes	—
337	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
338	C₂×Q₈	—	4	16	no	—
339	C₂×Q₈	—	4	8	no	—
340	C₂×Q₈	C₂²×C₄×C₆₇	1	16	yes	complemented
341	C₂×D₄	—	4	16	no	—
342	C₂×D₄	—	4	8	no	—
343	C₂×D₄	C₂²×C₄×C₆₇	1	16	yes	—
344	C₂²×C₄	—	2	4	no	—
345	C₂²×C₄	—	2	2	no	—
346	C₂²×C₄	—	2	4	no	—
347	C₂²×C₄	—	2	4	no	—
348	C₂²×C₄	—	2	4	no	—
349	C₂²×C₄	—	2	4	no	—
350	C₂²×C₄	—	2	4	no	—
351	C₂²×C₄	—	2	2	no	—
352	C₂²×C₄	—	2	4	no	—
353	C₂²×C₄	—	2	4	no	—
354	C₂²×C₄	—	2	4	no	—
355	C₂²×C₄	—	2	4	no	—
356	C₂²×C₄	—	2	4	no	—
357	C₂²×C₄	—	2	4	no	—
358	C₂²×C₄	—	2	4	no	—
359	C₂²×C₄	—	2	4	no	—
360	C₂²×C₄	—	2	4	no	—
361	C₂²×C₄	—	2	16	no	—
362	C₂²×C₄	—	4	8	no	—
363	C₂²×C₄	—	4	8	no	—
364	C₂²×C₄	—	4	32	no	—
365	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
366	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
367	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
368	C₂²×C₄	C₂²×C₄×C₆₇	1	8	yes	—
369	C₂²×C₄	#41×C₆₇	1	8	yes	—
370	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
371	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
372	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
373	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
374	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
375	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
376	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
377	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
378	C₄⋊C₄	—	2	16	no	—
379	C₄⋊C₄	—	4	8	no	—
380	C₂²⋊C₄	—	2	16	no	—
381	C₂²⋊C₄	—	4	8	no	—
382	C₂²⋊C₄	—	4	8	no	—
383	C₂²⋊C₄	C₂²×C₄×C₆₇	1	32	yes	complemented
384	C₂²⋊C₄	C₂²×C₄×C₆₇	1	32	yes	complemented
385	C₄²	—	4	4	no	—
386	C₄²	—	4	4	no	—
387	C₄²	—	4	8	no	—
388	C₄²	C₂²×C₄×C₆₇	1	16	yes	complemented
389	C₂³	—	2	16	no	—
390	C₂³	—	2	16	no	—
391	C₂³	—	2	16	no	—
392	C₂³	—	2	4	no	—
393	C₂³	—	2	4	no	—
394	C₂³	—	2	4	no	—
395	C₂³	—	2	4	no	—
396	C₂³	—	2	4	no	—
397	C₂³	—	2	4	no	—
398	C₂³	—	4	8	no	—
399	C₂³	—	4	8	no	—
400	C₂³	—	4	8	no	—
401	C₂³	—	4	8	no	—
402	C₂³	C₂³×C₄×C₆₇	1	8	yes	—
403	C₂³	C₂³×C₄×C₆₇	1	2	yes	—
404	C₂³	C₂×#41×C₆₇	1	4	yes	—
405	C₂³	H(5,𝔽₂)×C₆₇	1	4	yes	—
406	C₂³	H(5,𝔽₂)×C₆₇	1	2	yes	—
407	C₂³	2₋¹⁺⁴×C₆₇	1	2	yes	—
408	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
409	Q₈	—	4	16	no	—
410	D₄	—	4	16	no	—
411	C₂×C₄	—	2	8	no	—
412	C₂×C₄	—	2	8	no	—
413	C₂×C₄	—	2	16	no	—
414	C₂×C₄	—	2	8	no	—
415	C₂×C₄	—	2	16	no	—
416	C₂×C₄	—	2	4	no	—
417	C₂×C₄	—	2	2	no	—
418	C₂×C₄	—	2	4	no	—
419	C₂×C₄	—	2	4	no	—
420	C₂×C₄	—	2	4	no	—
421	C₂×C₄	—	2	4	no	—
422	C₂×C₄	—	2	4	no	—
423	C₂×C₄	—	2	2	no	—
424	C₂×C₄	—	2	4	no	—
425	C₂×C₄	—	2	4	no	—
426	C₂×C₄	—	2	4	no	—
427	C₂×C₄	—	2	4	no	—
428	C₂×C₄	—	2	4	no	—
429	C₂×C₄	—	2	4	no	—
430	C₂×C₄	—	2	4	no	—
431	C₂×C₄	—	2	4	no	—
432	C₂×C₄	—	2	4	no	—
433	C₂×C₄	—	4	32	no	—
434	C₂×C₄	—	4	8	no	—
435	C₂×C₄	—	4	8	no	—
436	C₂×C₄	—	4	8	no	—
437	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
438	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
439	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
440	C₂×C₄	C₂³×C₄×C₆₇	1	8	yes	—
441	C₂×C₄	C₂×#41×C₆₇	1	8	yes	—
442	C₂²	—	2	8	no	—
443	C₂²	—	2	8	no	—
444	C₂²	—	2	8	no	—
445	C₂²	—	4	32	no	—
446	C₂²	—	4	4	no	—
447	C₂²	—	4	4	no	—
448	C₂²	—	4	4	no	—
449	C₂²	—	4	4	no	—
450	C₂²	C₄×#41×C₆₇	1	4	yes	—
451	C₂²	#518×C₆₇	1	4	yes	—
452	C₂²	#520×C₆₇	1	4	yes	—
453	C₂²	#538×C₆₇	1	8	yes	—
454	C₂²	#542×C₆₇	1	8	yes	—
455	C₂²	C₂⁴×C₄×C₆₇	1	1	yes	characteristic
456	C₂²	C₂²×#41×C₆₇	1	2	yes	—
457	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	2	yes	—
458	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
459	C₂²	C₂×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
460	C₄	—	2	4	no	—
461	C₄	—	2	4	no	—
462	C₄	—	2	8	no	—
463	C₄	—	2	4	no	—
464	C₄	—	2	8	no	—
465	C₄	—	4	4	no	—
466	C₄	—	4	4	no	—
467	C₄	—	4	4	no	—
468	C₂	—	4	8	no	—
469	C₂	C₆₇×#2337	1	8	yes	direct product factor
470	C₂	C₂×C₄×#41×C₆₇	1	1	yes	characteristic
471	C₂	C₂×#518×C₆₇	1	1	yes	characteristic
472	C₂	C₂×#520×C₆₇	1	1	yes	characteristic
473	C₂	C₂×#538×C₆₇	1	2	yes	—
474	C₂	C₂×#542×C₆₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₆₇	C₂⁴
commutator	C₂²	C₂⁴×C₄×C₆₇
Frattini, cosocle	C₂³	C₂⁵×C₆₇
prime radical	C₂³×D₄×C₆₇	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×#2337	C₂	direct (N × H)
3. C₂×#391×C₆₇	C₂	semidirect
4. C₂×#388×C₆₇	C₂	semidirect
5. C₂×#388×C₆₇	C₂	semidirect
7. C₂×#385×C₆₇	C₂	semidirect
8. C₂×#383×C₆₇	C₂	semidirect
9. C₂×C₄×C₄⋊C₄×C₆₇	C₂	semidirect
10. C₂×C₄×C₂²⋊C₄×C₆₇	C₂	semidirect
14. C₂×#124×C₆₇	C₄	semidirect
27. #391×C₆₇	C₂²	semidirect
28. #388×C₆₇	C₂²	semidirect
29. #388×C₆₇	C₂²	semidirect
31. #385×C₆₇	C₂²	semidirect
32. #383×C₆₇	C₂²	semidirect
33. C₄×C₄⋊C₄×C₆₇	C₂²	semidirect
34. C₄×C₂²⋊C₄×C₆₇	C₂²	semidirect
44. C₂²×Q₈×C₆₇	C₂×C₄	semidirect
61. #124×C₆₇	C₂×C₄	semidirect
69. C₂×C₂²⋊C₄×C₆₇	C₂×C₄	semidirect
70. C₂×C₂²⋊C₄×C₆₇	C₂×C₄	semidirect
79. C₂×C₄²×C₆₇	C₂×C₄	semidirect
102. C₂×Q₈×C₆₇	C₂²×C₄	semidirect
145. C₂²⋊C₄×C₆₇	C₂²×C₄	semidirect
146. C₂²⋊C₄×C₆₇	C₂²×C₄	semidirect
150. C₄²×C₆₇	C₂²×C₄	semidirect
230. C₂×#2337	C₆₇	direct (N × H)
232. C₂×C₆₇	#2337	direct (N × H)
239. #2337	C₂×C₆₇	direct (N × H)
240. C₂×#391	C₂×C₆₇	semidirect
241. C₂×#388	C₂×C₆₇	semidirect
242. C₂×#388	C₂×C₆₇	semidirect
244. C₂×#385	C₂×C₆₇	semidirect
245. C₂×#383	C₂×C₆₇	semidirect
246. C₂×C₄×C₄⋊C₄	C₂×C₆₇	semidirect
247. C₂×C₄×C₂²⋊C₄	C₂×C₆₇	semidirect
248. C₆₇	C₂×#2337	direct (N × H)
252. C₂×#124	C₄×C₆₇	semidirect
265. #391	C₂²×C₆₇	semidirect
266. #388	C₂²×C₆₇	semidirect
267. #388	C₂²×C₆₇	semidirect
269. #385	C₂²×C₆₇	semidirect
270. #383	C₂²×C₆₇	semidirect
271. C₄×C₄⋊C₄	C₂²×C₆₇	semidirect
272. C₄×C₂²⋊C₄	C₂²×C₆₇	semidirect
282. C₂²×Q₈	C₂×C₄×C₆₇	semidirect
299. #124	C₂×C₄×C₆₇	semidirect
307. C₂×C₂²⋊C₄	C₂×C₄×C₆₇	semidirect
308. C₂×C₂²⋊C₄	C₂×C₄×C₆₇	semidirect
317. C₂×C₄²	C₂×C₄×C₆₇	semidirect
340. C₂×Q₈	C₂²×C₄×C₆₇	semidirect
383. C₂²⋊C₄	C₂²×C₄×C₆₇	semidirect
384. C₂²⋊C₄	C₂²×C₄×C₆₇	semidirect
388. C₄²	C₂²×C₄×C₆₇	semidirect
469. C₂	C₆₇×#2337	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
C₆₇×#2337	2
C₂×#391×C₆₇	2
C₂×#388×C₆₇	2
C₂×#386×C₆₇	2
C₂×#385×C₆₇	2
C₂×#383×C₆₇	2
C₂×C₄×C₄⋊C₄×C₆₇	2
C₂×C₄×C₂²⋊C₄×C₆₇	2
C₂×#2337	67

Subgroups — C2×C67×#2337 #30713331

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#2337 #30713331