Subgroups — C₂×C₆₇×#2624 #30713859

Order 17152.

3110 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#121×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×#117×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2624	C₂	1	16	yes	maximal, direct product factor
4	C₂×#399×C₆₇	C₂	1	2	yes	maximal, complemented
5	C₂×#397×C₆₇	C₂	1	2	yes	maximal, complemented
6	C₂×#394×C₆₇	C₂	1	4	yes	maximal, complemented
7	C₂×#390×C₆₇	C₂	1	2	yes	maximal, complemented
8	C₂×#389×C₆₇	C₂	1	2	yes	maximal, complemented
9	C₂×#380×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂³×D₄×C₆₇	C₂²	1	1	yes	characteristic
11	C₂³×D₄×C₆₇	C₂²	1	2	yes	—
12	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×#121×C₆₇	—	2	8	no	—
14	C₂×#121×C₆₇	C₂²	1	8	yes	—
15	C₂×#117×C₆₇	—	2	8	no	—
16	C₂×#117×C₆₇	C₂²	1	8	yes	complemented
17	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
18	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
19	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
20	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
21	C₂²×C₄⋊C₄×C₆₇	C₂²	1	4	yes	—
22	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
23	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
24	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
25	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
26	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
27	C₂²×C₄²×C₆₇	C₂²	1	2	yes	—
28	#399×C₆₇	C₂²	1	16	yes	complemented
29	#397×C₆₇	C₂²	1	16	yes	complemented
30	#394×C₆₇	C₂²	1	32	yes	complemented
31	#390×C₆₇	C₂²	1	16	yes	complemented
32	#389×C₆₇	C₂²	1	16	yes	complemented
33	#380×C₆₇	C₂²	1	8	yes	complemented
34	C₂×#95×C₆₇	C₂²	1	2	yes	—
35	C₂×#95×C₆₇	C₂²	1	4	yes	—
36	C₂⁵×C₆₇	C₂³	1	2	yes	—
37	C₂⁵×C₆₇	C₂³	1	1	yes	characteristic
38	C₂²×D₄×C₆₇	—	2	8	no	—
39	C₂²×D₄×C₆₇	—	2	4	no	—
40	C₂²×D₄×C₆₇	—	2	8	no	—
41	C₂²×D₄×C₆₇	—	4	8	no	—
42	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
43	C₂²×D₄×C₆₇	C₂³	1	8	yes	—
44	C₂³×C₄×C₆₇	—	2	8	no	—
45	C₂³×C₄×C₆₇	D₄	1	8	yes	—
46	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
47	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
48	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
49	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
50	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
51	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
52	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
53	#121×C₆₇	—	2	32	no	—
54	#117×C₆₇	—	2	32	no	—
55	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
56	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
57	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
58	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
59	C₂×C₄⋊C₄×C₆₇	—	4	8	no	—
60	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
61	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
62	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
63	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
64	C₂×C₄⋊C₄×C₆₇	C₂³	1	16	yes	—
65	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
66	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
67	C₂×C₂²⋊C₄×C₆₇	—	4	4	no	—
68	C₂×C₂²⋊C₄×C₆₇	—	4	8	no	—
69	C₂×C₂²⋊C₄×C₆₇	—	4	8	no	—
70	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
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	16	yes	—
74	C₂×C₂²⋊C₄×C₆₇	C₂³	1	16	yes	—
75	C₂×C₄²×C₆₇	—	2	8	no	—
76	C₂×C₄²×C₆₇	C₂³	1	8	yes	—
77	#95×C₆₇	C₂³	1	8	yes	—
78	#95×C₆₇	C₂³	1	16	yes	—
79	C₂⁴×C₆₇	—	2	4	no	—
80	C₂⁴×C₆₇	—	2	2	no	—
81	C₂⁴×C₆₇	—	2	4	no	—
82	C₂⁴×C₆₇	—	2	2	no	—
83	C₂⁴×C₆₇	—	2	4	no	—
84	C₂⁴×C₆₇	—	2	8	no	—
85	C₂⁴×C₆₇	—	2	4	no	—
86	C₂⁴×C₆₇	—	2	8	no	—
87	C₂⁴×C₆₇	—	2	4	no	—
88	C₂⁴×C₆₇	C₂×D₄	1	4	yes	—
89	C₂⁴×C₆₇	C₂⁴	1	4	yes	—
90	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
91	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
92	C₂×D₄×C₆₇	—	2	16	no	—
93	C₂×D₄×C₆₇	—	2	8	no	—
94	C₂×D₄×C₆₇	—	2	16	no	—
95	C₂×D₄×C₆₇	—	4	16	no	—
96	C₂×D₄×C₆₇	—	4	32	no	—
97	C₂×D₄×C₆₇	—	4	16	no	—
98	C₂²×C₄×C₆₇	—	2	4	no	—
99	C₂²×C₄×C₆₇	—	2	32	no	—
100	C₂²×C₄×C₆₇	—	2	16	no	—
101	C₂²×C₄×C₆₇	—	2	4	no	—
102	C₂²×C₄×C₆₇	—	2	4	no	—
103	C₂²×C₄×C₆₇	—	2	4	no	—
104	C₂²×C₄×C₆₇	—	2	8	no	—
105	C₂²×C₄×C₆₇	—	2	4	no	—
106	C₂²×C₄×C₆₇	—	2	4	no	—
107	C₂²×C₄×C₆₇	—	2	8	no	—
108	C₂²×C₄×C₆₇	—	2	8	no	—
109	C₂²×C₄×C₆₇	—	2	4	no	—
110	C₂²×C₄×C₆₇	—	2	8	no	—
111	C₂²×C₄×C₆₇	—	2	4	no	—
112	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
113	C₂²×C₄×C₆₇	C₂×D₄	1	16	yes	—
114	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
115	C₂²×C₄×C₆₇	#41	1	8	yes	—
116	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
117	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
118	C₂²×C₄×C₆₇	C₂⁴	1	8	yes	—
119	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
120	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
121	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
122	C₄⋊C₄×C₆₇	—	2	8	no	—
123	C₄⋊C₄×C₆₇	—	2	8	no	—
124	C₄⋊C₄×C₆₇	—	2	8	no	—
125	C₄⋊C₄×C₆₇	—	2	16	no	—
126	C₄⋊C₄×C₆₇	—	4	16	no	—
127	C₂²⋊C₄×C₆₇	—	2	16	no	—
128	C₂²⋊C₄×C₆₇	—	2	16	no	—
129	C₂²⋊C₄×C₆₇	—	4	8	no	—
130	C₂²⋊C₄×C₆₇	—	4	16	no	—
131	C₂²⋊C₄×C₆₇	—	4	16	no	—
132	C₄²×C₆₇	—	2	16	no	—
133	C₂³×C₆₇	—	2	8	no	—
134	C₂³×C₆₇	—	2	8	no	—
135	C₂³×C₆₇	—	2	8	no	—
136	C₂³×C₆₇	—	2	4	no	—
137	C₂³×C₆₇	—	2	2	no	—
138	C₂³×C₆₇	—	2	4	no	—
139	C₂³×C₆₇	—	2	2	no	—
140	C₂³×C₆₇	—	2	4	no	—
141	C₂³×C₆₇	—	2	8	no	—
142	C₂³×C₆₇	—	2	4	no	—
143	C₂³×C₆₇	—	2	8	no	—
144	C₂³×C₆₇	—	2	4	no	—
145	C₂³×C₆₇	—	4	8	no	—
146	C₂³×C₆₇	—	4	8	no	—
147	C₂³×C₆₇	—	4	8	no	—
148	C₂³×C₆₇	—	4	8	no	—
149	C₂³×C₆₇	—	4	8	no	—
150	C₂³×C₆₇	—	4	16	no	—
151	C₂³×C₆₇	—	4	16	no	—
152	C₂³×C₆₇	C₂²×D₄	1	4	yes	—
153	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
154	C₂³×C₆₇	C₂×#41	1	4	yes	—
155	C₂³×C₆₇	H(5,𝔽₂)	1	4	yes	—
156	C₂³×C₆₇	H(5,𝔽₂)	1	2	yes	—
157	C₂³×C₆₇	2₋¹⁺⁴	1	2	yes	—
158	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
159	D₄×C₆₇	—	4	32	no	—
160	D₄×C₆₇	—	4	16	no	—
161	C₂×C₄×C₆₇	—	2	8	no	—
162	C₂×C₄×C₆₇	—	2	4	no	—
163	C₂×C₄×C₆₇	—	2	32	no	—
164	C₂×C₄×C₆₇	—	2	4	no	—
165	C₂×C₄×C₆₇	—	2	4	no	—
166	C₂×C₄×C₆₇	—	2	4	no	—
167	C₂×C₄×C₆₇	—	2	8	no	—
168	C₂×C₄×C₆₇	—	2	4	no	—
169	C₂×C₄×C₆₇	—	2	4	no	—
170	C₂×C₄×C₆₇	—	2	8	no	—
171	C₂×C₄×C₆₇	—	2	8	no	—
172	C₂×C₄×C₆₇	—	2	4	no	—
173	C₂×C₄×C₆₇	—	2	8	no	—
174	C₂×C₄×C₆₇	—	2	4	no	—
175	C₂×C₄×C₆₇	—	4	8	no	—
176	C₂×C₄×C₆₇	—	4	8	no	—
177	C₂×C₄×C₆₇	—	4	8	no	—
178	C₂×C₄×C₆₇	—	4	16	no	—
179	C₂×C₄×C₆₇	#123	1	16	yes	—
180	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
181	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
182	C₂×C₄×C₆₇	C₂×#41	1	8	yes	—
183	C₂²×C₆₇	—	2	4	no	—
184	C₂²×C₆₇	—	2	4	no	—
185	C₂²×C₆₇	—	2	4	no	—
186	C₂²×C₆₇	—	4	16	no	—
187	C₂²×C₆₇	—	4	4	no	—
188	C₂²×C₆₇	—	4	4	no	—
189	C₂²×C₆₇	—	4	4	no	—
190	C₂²×C₆₇	—	4	4	no	—
191	C₂²×C₆₇	—	4	4	no	—
192	C₂²×C₆₇	—	4	8	no	—
193	C₂²×C₆₇	—	4	8	no	—
194	C₂²×C₆₇	—	8	16	no	—
195	C₂²×C₆₇	C₂×#123	1	4	yes	—
196	C₂²×C₆₇	#535	1	4	yes	—
197	C₂²×C₆₇	#537	1	4	yes	—
198	C₂²×C₆₇	#540	1	8	yes	—
199	C₂²×C₆₇	#542	1	8	yes	—
200	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
201	C₂²×C₆₇	C₂²×#41	1	2	yes	—
202	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	2	yes	—
203	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
204	C₂²×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
205	C₄×C₆₇	—	2	4	no	—
206	C₄×C₆₇	—	4	4	no	—
207	C₄×C₆₇	—	4	4	no	—
208	C₄×C₆₇	—	4	4	no	—
209	C₄×C₆₇	—	4	8	no	—
210	C₄×C₆₇	C₂×#123	1	8	yes	—
211	C₂×#2624	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
212	C₂×C₆₇	—	4	4	no	—
213	C₂×C₆₇	—	8	4	no	—
214	C₂×C₆₇	#2624	1	8	yes	direct product factor
215	C₂×C₆₇	C₂²×#123	1	1	yes	characteristic
216	C₂×C₆₇	C₂×#535	1	1	yes	characteristic
217	C₂×C₆₇	C₂×#537	1	1	yes	characteristic
218	C₂×C₆₇	C₂×#540	1	2	yes	—
219	C₂×C₆₇	C₂×#542	1	2	yes	—
220	C₂²×#121	C₂×C₆₇	1	1	yes	characteristic
221	C₂²×#117	C₂×C₆₇	1	1	yes	characteristic, complemented
222	#2624	C₂×C₆₇	1	16	yes	direct product factor
223	C₂×#399	C₂×C₆₇	1	2	yes	complemented
224	C₂×#397	C₂×C₆₇	1	2	yes	complemented
225	C₂×#394	C₂×C₆₇	1	4	yes	complemented
226	C₂×#390	C₂×C₆₇	1	2	yes	complemented
227	C₂×#389	C₂×C₆₇	1	2	yes	complemented
228	C₂×#380	C₂×C₆₇	1	1	yes	characteristic, complemented
229	C₆₇	C₂×#2624	1	1	yes	characteristic, direct product factor, Sylow
230	C₂³×D₄	C₂²×C₆₇	1	1	yes	characteristic
231	C₂³×D₄	C₂²×C₆₇	1	2	yes	—
232	C₂⁴×C₄	C₂²×C₆₇	1	1	yes	characteristic
233	C₂×#121	—	2	8	no	—
234	C₂×#121	C₂²×C₆₇	1	8	yes	—
235	C₂×#117	—	2	8	no	—
236	C₂×#117	C₂²×C₆₇	1	8	yes	complemented
237	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
238	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
239	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
240	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
241	C₂²×C₄⋊C₄	C₂²×C₆₇	1	4	yes	—
242	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
243	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
244	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
245	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
246	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
247	C₂²×C₄²	C₂²×C₆₇	1	2	yes	—
248	#399	C₂²×C₆₇	1	16	yes	complemented
249	#397	C₂²×C₆₇	1	16	yes	complemented
250	#394	C₂²×C₆₇	1	32	yes	complemented
251	#390	C₂²×C₆₇	1	16	yes	complemented
252	#389	C₂²×C₆₇	1	16	yes	complemented
253	#380	C₂²×C₆₇	1	8	yes	complemented
254	C₂×#95	C₂²×C₆₇	1	4	yes	—
255	C₂×#95	C₂²×C₆₇	1	2	yes	—
256	C₂⁵	C₂³×C₆₇	1	2	yes	—
257	C₂⁵	C₂³×C₆₇	1	1	yes	characteristic
258	C₂²×D₄	—	2	8	no	—
259	C₂²×D₄	—	2	4	no	—
260	C₂²×D₄	—	2	8	no	—
261	C₂²×D₄	—	4	8	no	—
262	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
263	C₂²×D₄	C₂³×C₆₇	1	8	yes	—
264	C₂³×C₄	—	2	8	no	—
265	C₂³×C₄	D₄×C₆₇	1	8	yes	—
266	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
267	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
268	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
269	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
270	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
271	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
272	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
273	#121	—	2	32	no	—
274	#117	—	2	32	no	—
275	C₂×C₄⋊C₄	—	2	4	no	—
276	C₂×C₄⋊C₄	—	2	4	no	—
277	C₂×C₄⋊C₄	—	2	4	no	—
278	C₂×C₄⋊C₄	—	2	8	no	—
279	C₂×C₄⋊C₄	—	4	8	no	—
280	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
281	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
282	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
283	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
284	C₂×C₄⋊C₄	C₂³×C₆₇	1	16	yes	—
285	C₂×C₂²⋊C₄	—	2	8	no	—
286	C₂×C₂²⋊C₄	—	2	8	no	—
287	C₂×C₂²⋊C₄	—	4	4	no	—
288	C₂×C₂²⋊C₄	—	4	8	no	—
289	C₂×C₂²⋊C₄	—	4	8	no	—
290	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
291	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
292	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
293	C₂×C₂²⋊C₄	C₂³×C₆₇	1	16	yes	—
294	C₂×C₂²⋊C₄	C₂³×C₆₇	1	16	yes	—
295	C₂×C₄²	—	2	8	no	—
296	C₂×C₄²	C₂³×C₆₇	1	8	yes	—
297	#95	C₂³×C₆₇	1	8	yes	—
298	#95	C₂³×C₆₇	1	16	yes	—
299	C₂⁴	—	2	4	no	—
300	C₂⁴	—	2	2	no	—
301	C₂⁴	—	2	4	no	—
302	C₂⁴	—	2	2	no	—
303	C₂⁴	—	2	4	no	—
304	C₂⁴	—	2	8	no	—
305	C₂⁴	—	2	4	no	—
306	C₂⁴	—	2	8	no	—
307	C₂⁴	—	2	4	no	—
308	C₂⁴	C₂×D₄×C₆₇	1	4	yes	—
309	C₂⁴	C₂⁴×C₆₇	1	4	yes	—
310	C₂⁴	C₂⁴×C₆₇	1	2	yes	—
311	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
312	C₂×D₄	—	2	16	no	—
313	C₂×D₄	—	2	8	no	—
314	C₂×D₄	—	2	16	no	—
315	C₂×D₄	—	4	16	no	—
316	C₂×D₄	—	4	32	no	—
317	C₂×D₄	—	4	16	no	—
318	C₂²×C₄	—	2	4	no	—
319	C₂²×C₄	—	2	32	no	—
320	C₂²×C₄	—	2	16	no	—
321	C₂²×C₄	—	2	4	no	—
322	C₂²×C₄	—	2	4	no	—
323	C₂²×C₄	—	2	4	no	—
324	C₂²×C₄	—	2	8	no	—
325	C₂²×C₄	—	2	4	no	—
326	C₂²×C₄	—	2	4	no	—
327	C₂²×C₄	—	2	8	no	—
328	C₂²×C₄	—	2	8	no	—
329	C₂²×C₄	—	2	4	no	—
330	C₂²×C₄	—	2	8	no	—
331	C₂²×C₄	—	2	4	no	—
332	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
333	C₂²×C₄	C₂×D₄×C₆₇	1	16	yes	—
334	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
335	C₂²×C₄	#41×C₆₇	1	8	yes	—
336	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
337	C₂²×C₄	C₂⁴×C₆₇	1	8	yes	—
338	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
339	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
340	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
341	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
342	C₄⋊C₄	—	2	8	no	—
343	C₄⋊C₄	—	2	8	no	—
344	C₄⋊C₄	—	2	8	no	—
345	C₄⋊C₄	—	2	16	no	—
346	C₄⋊C₄	—	4	16	no	—
347	C₂²⋊C₄	—	2	16	no	—
348	C₂²⋊C₄	—	2	16	no	—
349	C₂²⋊C₄	—	4	8	no	—
350	C₂²⋊C₄	—	4	16	no	—
351	C₂²⋊C₄	—	4	16	no	—
352	C₄²	—	2	16	no	—
353	C₂³	—	2	8	no	—
354	C₂³	—	2	8	no	—
355	C₂³	—	2	8	no	—
356	C₂³	—	2	4	no	—
357	C₂³	—	2	2	no	—
358	C₂³	—	2	4	no	—
359	C₂³	—	2	2	no	—
360	C₂³	—	2	4	no	—
361	C₂³	—	2	8	no	—
362	C₂³	—	2	4	no	—
363	C₂³	—	2	8	no	—
364	C₂³	—	2	4	no	—
365	C₂³	—	4	8	no	—
366	C₂³	—	4	8	no	—
367	C₂³	—	4	8	no	—
368	C₂³	—	4	8	no	—
369	C₂³	—	4	8	no	—
370	C₂³	—	4	16	no	—
371	C₂³	—	4	16	no	—
372	C₂³	C₂²×D₄×C₆₇	1	4	yes	—
373	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
374	C₂³	C₂×#41×C₆₇	1	4	yes	—
375	C₂³	H(5,𝔽₂)×C₆₇	1	4	yes	—
376	C₂³	H(5,𝔽₂)×C₆₇	1	2	yes	—
377	C₂³	2₋¹⁺⁴×C₆₇	1	2	yes	—
378	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
379	D₄	—	4	32	no	—
380	D₄	—	4	16	no	—
381	C₂×C₄	—	2	8	no	—
382	C₂×C₄	—	2	4	no	—
383	C₂×C₄	—	2	32	no	—
384	C₂×C₄	—	2	4	no	—
385	C₂×C₄	—	2	4	no	—
386	C₂×C₄	—	2	4	no	—
387	C₂×C₄	—	2	8	no	—
388	C₂×C₄	—	2	4	no	—
389	C₂×C₄	—	2	4	no	—
390	C₂×C₄	—	2	8	no	—
391	C₂×C₄	—	2	8	no	—
392	C₂×C₄	—	2	4	no	—
393	C₂×C₄	—	2	8	no	—
394	C₂×C₄	—	2	4	no	—
395	C₂×C₄	—	4	8	no	—
396	C₂×C₄	—	4	8	no	—
397	C₂×C₄	—	4	8	no	—
398	C₂×C₄	—	4	16	no	—
399	C₂×C₄	#123×C₆₇	1	16	yes	—
400	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
401	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
402	C₂×C₄	C₂×#41×C₆₇	1	8	yes	—
403	C₂²	—	2	4	no	—
404	C₂²	—	2	4	no	—
405	C₂²	—	2	4	no	—
406	C₂²	—	4	16	no	—
407	C₂²	—	4	4	no	—
408	C₂²	—	4	4	no	—
409	C₂²	—	4	4	no	—
410	C₂²	—	4	4	no	—
411	C₂²	—	4	4	no	—
412	C₂²	—	4	8	no	—
413	C₂²	—	4	8	no	—
414	C₂²	—	8	16	no	—
415	C₂²	C₂×#123×C₆₇	1	4	yes	—
416	C₂²	#535×C₆₇	1	4	yes	—
417	C₂²	#537×C₆₇	1	4	yes	—
418	C₂²	#540×C₆₇	1	8	yes	—
419	C₂²	#542×C₆₇	1	8	yes	—
420	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
421	C₂²	C₂²×#41×C₆₇	1	2	yes	—
422	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	2	yes	—
423	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
424	C₂²	C₂×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
425	C₄	—	2	4	no	—
426	C₄	—	4	4	no	—
427	C₄	—	4	4	no	—
428	C₄	—	4	4	no	—
429	C₄	—	4	8	no	—
430	C₄	C₂×#123×C₆₇	1	8	yes	—
431	C₂	—	4	4	no	—
432	C₂	—	8	4	no	—
433	C₂	C₆₇×#2624	1	8	yes	direct product factor
434	C₂	C₂²×#123×C₆₇	1	1	yes	characteristic
435	C₂	C₂×#535×C₆₇	1	1	yes	characteristic
436	C₂	C₂×#537×C₆₇	1	1	yes	characteristic
437	C₂	C₂×#540×C₆₇	1	2	yes	—
438	C₂	C₂×#542×C₆₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₆₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₆₇
prime radical	C₂²×#121×C₆₇	C₂
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
2. C₂²×#117×C₆₇	C₂	semidirect
3. C₆₇×#2624	C₂	direct (N × H)
4. C₂×#399×C₆₇	C₂	semidirect
5. C₂×#397×C₆₇	C₂	semidirect
6. C₂×#394×C₆₇	C₂	semidirect
7. C₂×#390×C₆₇	C₂	semidirect
8. C₂×#389×C₆₇	C₂	semidirect
9. C₂×#380×C₆₇	C₂	semidirect
16. C₂×#117×C₆₇	C₂²	semidirect
28. #399×C₆₇	C₂²	semidirect
29. #397×C₆₇	C₂²	semidirect
30. #394×C₆₇	C₂²	semidirect
31. #390×C₆₇	C₂²	semidirect
32. #389×C₆₇	C₂²	semidirect
33. #380×C₆₇	C₂²	semidirect
211. C₂×#2624	C₆₇	direct (N × H)
214. C₂×C₆₇	#2624	direct (N × H)
221. C₂²×#117	C₂×C₆₇	semidirect
222. #2624	C₂×C₆₇	direct (N × H)
223. C₂×#399	C₂×C₆₇	semidirect
224. C₂×#397	C₂×C₆₇	semidirect
225. C₂×#394	C₂×C₆₇	semidirect
226. C₂×#390	C₂×C₆₇	semidirect
227. C₂×#389	C₂×C₆₇	semidirect
228. C₂×#380	C₂×C₆₇	semidirect
229. C₆₇	C₂×#2624	direct (N × H)
236. C₂×#117	C₂²×C₆₇	semidirect
248. #399	C₂²×C₆₇	semidirect
249. #397	C₂²×C₆₇	semidirect
250. #394	C₂²×C₆₇	semidirect
251. #390	C₂²×C₆₇	semidirect
252. #389	C₂²×C₆₇	semidirect
253. #380	C₂²×C₆₇	semidirect
433. C₂	C₆₇×#2624	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×#121×C₆₇	2
C₂²×#117×C₆₇	2
C₆₇×#2624	2
C₂×#399×C₆₇	2
C₂×#397×C₆₇	2
C₂×#394×C₆₇	2
C₂×#390×C₆₇	2
C₂×#389×C₆₇	2
C₂×#380×C₆₇	2
C₂×#2624	67

Subgroups — C2×C67×#2624 #30713859

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#2624 #30713859