Subgroups — C₂×C₆₇×#2832 #30715916

Order 17152.

2238 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#2832	C₂	1	16	yes	maximal, direct product factor
2	C₂×#400×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₂×#397×C₆₇	C₂	1	2	yes	maximal, complemented
4	C₂×#395×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#394×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#394×C₆₇	C₂	1	2	yes	maximal
7	C₂×#393×C₆₇	C₂	1	2	yes	maximal, complemented
8	C₂×#393×C₆₇	C₂	1	2	yes	maximal, complemented
9	C₂×#391×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#388×C₆₇	C₂	1	2	yes	maximal, complemented
11	C₂³×Q₈×C₆₇	C₂²	1	1	yes	characteristic
12	C₂³×Q₈×C₆₇	C₂²	1	1	yes	characteristic
13	C₂³×D₄×C₆₇	C₂²	1	1	yes	characteristic
14	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
15	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
16	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
17	C₂²×C₂²⋊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	#400×C₆₇	C₂²	1	16	yes	complemented
25	#397×C₆₇	C₂²	1	16	yes	complemented
26	#395×C₆₇	C₂²	1	8	yes	complemented
27	#394×C₆₇	C₂²	1	8	yes	—
28	#394×C₆₇	C₂²	1	16	yes	—
29	#393×C₆₇	C₂²	1	16	yes	complemented
30	#393×C₆₇	C₂²	1	16	yes	complemented
31	#391×C₆₇	C₂²	1	8	yes	complemented
32	#388×C₆₇	C₂²	1	16	yes	complemented
33	C₂×#95×C₆₇	C₂²	1	2	yes	—
34	C₂×#95×C₆₇	C₂²	1	2	yes	—
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	2	yes	—
39	C₂×#95×C₆₇	C₂²	1	1	yes	characteristic
40	C₂×#95×C₆₇	C₂²	1	1	yes	characteristic
41	C₂⁵×C₆₇	C₂³	1	2	yes	—
42	C₂²×Q₈×C₆₇	—	2	4	no	—
43	C₂²×Q₈×C₆₇	—	4	2	no	—
44	C₂²×Q₈×C₆₇	—	4	2	no	—
45	C₂²×Q₈×C₆₇	—	4	2	no	—
46	C₂²×Q₈×C₆₇	—	4	4	no	—
47	C₂²×Q₈×C₆₇	C₂³	1	4	yes	—
48	C₂²×Q₈×C₆₇	C₂³	1	4	yes	—
49	C₂²×D₄×C₆₇	—	4	2	no	—
50	C₂²×D₄×C₆₇	—	4	4	no	—
51	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
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	2	yes	—
57	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
58	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
59	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
60	C₂×C₄⋊C₄×C₆₇	—	4	2	no	—
61	C₂×C₄⋊C₄×C₆₇	—	4	4	no	—
62	C₂×C₄⋊C₄×C₆₇	—	4	4	no	—
63	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
64	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	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	—	4	4	no	—
70	C₂×C₂²⋊C₄×C₆₇	—	4	4	no	—
71	C₂×C₂²⋊C₄×C₆₇	—	4	4	no	—
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×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
77	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
78	C₂×C₄²×C₆₇	—	4	2	no	—
79	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
80	#95×C₆₇	C₂³	1	8	yes	—
81	#95×C₆₇	C₂³	1	8	yes	—
82	#95×C₆₇	C₂³	1	8	yes	—
83	#95×C₆₇	C₂³	1	8	yes	—
84	#95×C₆₇	C₂³	1	8	yes	—
85	#95×C₆₇	C₂³	1	8	yes	—
86	#95×C₆₇	C₂³	1	4	yes	—
87	#95×C₆₇	C₂³	1	4	yes	—
88	C₂⁴×C₆₇	—	2	4	no	—
89	C₂⁴×C₆₇	—	2	4	no	—
90	C₂⁴×C₆₇	—	2	4	no	—
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	C₂⁴	1	4	yes	—
96	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
97	C₂×Q₈×C₆₇	—	2	8	no	—
98	C₂×Q₈×C₆₇	—	4	4	no	—
99	C₂×Q₈×C₆₇	—	4	4	no	—
100	C₂×Q₈×C₆₇	—	4	4	no	—
101	C₂×Q₈×C₆₇	—	4	8	no	—
102	C₂×Q₈×C₆₇	—	4	16	no	—
103	C₂×Q₈×C₆₇	—	4	16	no	—
104	C₂×D₄×C₆₇	—	4	4	no	—
105	C₂×D₄×C₆₇	—	4	8	no	—
106	C₂×D₄×C₆₇	—	4	16	no	—
107	C₂²×C₄×C₆₇	—	2	2	no	—
108	C₂²×C₄×C₆₇	—	2	2	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	4	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	2	no	—
122	C₂²×C₄×C₆₇	—	2	4	no	—
123	C₂²×C₄×C₆₇	—	2	2	no	—
124	C₂²×C₄×C₆₇	—	2	4	no	—
125	C₂²×C₄×C₆₇	—	2	4	no	—
126	C₂²×C₄×C₆₇	—	2	4	no	—
127	C₂²×C₄×C₆₇	—	2	4	no	—
128	C₂²×C₄×C₆₇	#41	1	4	yes	—
129	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
130	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
131	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
132	C₂²×C₄×C₆₇	C₂⁴	1	2	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	C₄⋊C₄×C₆₇	—	4	4	no	—
138	C₄⋊C₄×C₆₇	—	4	8	no	—
139	C₄⋊C₄×C₆₇	—	4	8	no	—
140	C₂²⋊C₄×C₆₇	—	2	16	no	—
141	C₂²⋊C₄×C₆₇	—	4	8	no	—
142	C₂²⋊C₄×C₆₇	—	4	8	no	—
143	C₂²⋊C₄×C₆₇	—	4	8	no	—
144	C₂²⋊C₄×C₆₇	—	4	8	no	—
145	C₂²⋊C₄×C₆₇	—	4	8	no	—
146	C₄²×C₆₇	—	4	4	no	—
147	C₂³×C₆₇	—	2	4	no	—
148	C₂³×C₆₇	—	2	4	no	—
149	C₂³×C₆₇	—	2	4	no	—
150	C₂³×C₆₇	—	2	4	no	—
151	C₂³×C₆₇	—	2	4	no	—
152	C₂³×C₆₇	—	2	4	no	—
153	C₂³×C₆₇	—	2	4	no	—
154	C₂³×C₆₇	—	4	8	no	—
155	C₂³×C₆₇	—	4	8	no	—
156	C₂³×C₆₇	—	4	8	no	—
157	C₂³×C₆₇	—	4	8	no	—
158	C₂³×C₆₇	—	4	8	no	—
159	C₂³×C₆₇	—	4	8	no	—
160	C₂³×C₆₇	—	4	8	no	—
161	C₂³×C₆₇	C₂×#41	1	2	yes	—
162	C₂³×C₆₇	H(5,𝔽₂)	1	2	yes	—
163	C₂³×C₆₇	H(5,𝔽₂)	1	2	yes	—
164	C₂³×C₆₇	H(5,𝔽₂)	1	4	yes	—
165	C₂³×C₆₇	2₋¹⁺⁴	1	4	yes	—
166	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
167	Q₈×C₆₇	—	4	16	no	—
168	Q₈×C₆₇	—	4	16	no	—
169	D₄×C₆₇	—	4	16	no	—
170	C₂×C₄×C₆₇	—	2	8	no	—
171	C₂×C₄×C₆₇	—	2	2	no	—
172	C₂×C₄×C₆₇	—	2	2	no	—
173	C₂×C₄×C₆₇	—	2	4	no	—
174	C₂×C₄×C₆₇	—	2	4	no	—
175	C₂×C₄×C₆₇	—	2	4	no	—
176	C₂×C₄×C₆₇	—	2	4	no	—
177	C₂×C₄×C₆₇	—	2	4	no	—
178	C₂×C₄×C₆₇	—	2	4	no	—
179	C₂×C₄×C₆₇	—	2	4	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	2	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	—	4	8	no	—
193	C₂×C₄×C₆₇	—	4	8	no	—
194	C₂×C₄×C₆₇	—	4	4	no	—
195	C₂×C₄×C₆₇	—	4	8	no	—
196	C₂×C₄×C₆₇	—	4	4	no	—
197	C₂×C₄×C₆₇	—	4	8	no	—
198	C₂×C₄×C₆₇	—	4	8	no	—
199	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
200	C₂²×C₆₇	—	4	4	no	—
201	C₂²×C₆₇	—	4	4	no	—
202	C₂²×C₆₇	—	4	4	no	—
203	C₂²×C₆₇	—	4	4	no	—
204	C₂²×C₆₇	—	4	4	no	—
205	C₂²×C₆₇	—	4	4	no	—
206	C₂²×C₆₇	—	4	4	no	—
207	C₂²×C₆₇	—	8	16	no	—
208	C₂²×C₆₇	#538	1	4	yes	—
209	C₂²×C₆₇	#542	1	8	yes	—
210	C₂²×C₆₇	H(3,𝔽₄)	1	4	yes	—
211	C₂²×C₆₇	#562	1	8	yes	—
212	C₂²×C₆₇	#563	1	4	yes	—
213	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
214	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
215	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
216	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	2	yes	—
217	C₂²×C₆₇	C₂×2₋¹⁺⁴	1	2	yes	—
218	C₄×C₆₇	—	2	4	no	—
219	C₄×C₆₇	—	4	4	no	—
220	C₄×C₆₇	—	4	4	no	—
221	C₄×C₆₇	—	4	2	no	—
222	C₄×C₆₇	—	4	4	no	—
223	C₄×C₆₇	—	4	2	no	—
224	C₄×C₆₇	—	4	4	no	—
225	C₄×C₆₇	—	4	4	no	—
226	C₂×#2832	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
227	C₂×C₆₇	—	8	4	no	—
228	C₂×C₆₇	#2832	1	8	yes	direct product factor
229	C₂×C₆₇	C₂×#538	1	1	yes	characteristic
230	C₂×C₆₇	C₂×#542	1	2	yes	—
231	C₂×C₆₇	C₂×H(3,𝔽₄)	1	1	yes	characteristic
232	C₂×C₆₇	C₂×#562	1	2	yes	—
233	C₂×C₆₇	C₂×#563	1	1	yes	characteristic
234	#2832	C₂×C₆₇	1	16	yes	direct product factor
235	C₂×#400	C₂×C₆₇	1	2	yes	complemented
236	C₂×#397	C₂×C₆₇	1	2	yes	complemented
237	C₂×#395	C₂×C₆₇	1	1	yes	characteristic, complemented
238	C₂×#394	C₂×C₆₇	1	1	yes	characteristic
239	C₂×#394	C₂×C₆₇	1	2	yes	—
240	C₂×#393	C₂×C₆₇	1	2	yes	complemented
241	C₂×#393	C₂×C₆₇	1	2	yes	complemented
242	C₂×#391	C₂×C₆₇	1	1	yes	characteristic, complemented
243	C₂×#388	C₂×C₆₇	1	2	yes	complemented
244	C₆₇	C₂×#2832	1	1	yes	characteristic, direct product factor, Sylow
245	C₂³×Q₈	C₂²×C₆₇	1	1	yes	characteristic
246	C₂³×Q₈	C₂²×C₆₇	1	1	yes	characteristic
247	C₂³×D₄	C₂²×C₆₇	1	1	yes	characteristic
248	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
249	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
250	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
251	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
252	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
253	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
254	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
255	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
256	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
257	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
258	#400	C₂²×C₆₇	1	16	yes	complemented
259	#397	C₂²×C₆₇	1	16	yes	complemented
260	#395	C₂²×C₆₇	1	8	yes	complemented
261	#394	C₂²×C₆₇	1	16	yes	—
262	#394	C₂²×C₆₇	1	8	yes	—
263	#393	C₂²×C₆₇	1	16	yes	complemented
264	#393	C₂²×C₆₇	1	16	yes	complemented
265	#391	C₂²×C₆₇	1	8	yes	complemented
266	#388	C₂²×C₆₇	1	16	yes	complemented
267	C₂×#95	C₂²×C₆₇	1	2	yes	—
268	C₂×#95	C₂²×C₆₇	1	2	yes	—
269	C₂×#95	C₂²×C₆₇	1	2	yes	—
270	C₂×#95	C₂²×C₆₇	1	2	yes	—
271	C₂×#95	C₂²×C₆₇	1	2	yes	—
272	C₂×#95	C₂²×C₆₇	1	1	yes	characteristic
273	C₂×#95	C₂²×C₆₇	1	2	yes	—
274	C₂×#95	C₂²×C₆₇	1	1	yes	characteristic
275	C₂⁵	C₂³×C₆₇	1	2	yes	—
276	C₂²×Q₈	—	2	4	no	—
277	C₂²×Q₈	—	4	2	no	—
278	C₂²×Q₈	—	4	2	no	—
279	C₂²×Q₈	—	4	2	no	—
280	C₂²×Q₈	—	4	4	no	—
281	C₂²×Q₈	C₂³×C₆₇	1	4	yes	—
282	C₂²×Q₈	C₂³×C₆₇	1	4	yes	—
283	C₂²×D₄	—	4	2	no	—
284	C₂²×D₄	—	4	4	no	—
285	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
286	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
287	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
288	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
289	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
290	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
291	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
292	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
293	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
294	C₂×C₄⋊C₄	—	4	2	no	—
295	C₂×C₄⋊C₄	—	4	4	no	—
296	C₂×C₄⋊C₄	—	4	4	no	—
297	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
298	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
299	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
300	C₂×C₂²⋊C₄	—	2	8	no	—
301	C₂×C₂²⋊C₄	—	4	4	no	—
302	C₂×C₂²⋊C₄	—	4	4	no	—
303	C₂×C₂²⋊C₄	—	4	4	no	—
304	C₂×C₂²⋊C₄	—	4	4	no	—
305	C₂×C₂²⋊C₄	—	4	4	no	—
306	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
307	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
308	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
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×C₄²	—	4	2	no	—
313	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
314	#95	C₂³×C₆₇	1	8	yes	—
315	#95	C₂³×C₆₇	1	8	yes	—
316	#95	C₂³×C₆₇	1	8	yes	—
317	#95	C₂³×C₆₇	1	8	yes	—
318	#95	C₂³×C₆₇	1	8	yes	—
319	#95	C₂³×C₆₇	1	8	yes	—
320	#95	C₂³×C₆₇	1	4	yes	—
321	#95	C₂³×C₆₇	1	4	yes	—
322	C₂⁴	—	2	4	no	—
323	C₂⁴	—	2	4	no	—
324	C₂⁴	—	2	4	no	—
325	C₂⁴	—	2	4	no	—
326	C₂⁴	—	2	4	no	—
327	C₂⁴	—	2	4	no	—
328	C₂⁴	—	2	4	no	—
329	C₂⁴	C₂⁴×C₆₇	1	4	yes	—
330	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
331	C₂×Q₈	—	2	8	no	—
332	C₂×Q₈	—	4	4	no	—
333	C₂×Q₈	—	4	4	no	—
334	C₂×Q₈	—	4	8	no	—
335	C₂×Q₈	—	4	4	no	—
336	C₂×Q₈	—	4	16	no	—
337	C₂×Q₈	—	4	16	no	—
338	C₂×D₄	—	4	8	no	—
339	C₂×D₄	—	4	4	no	—
340	C₂×D₄	—	4	16	no	—
341	C₂²×C₄	—	2	2	no	—
342	C₂²×C₄	—	2	2	no	—
343	C₂²×C₄	—	2	4	no	—
344	C₂²×C₄	—	2	4	no	—
345	C₂²×C₄	—	2	4	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	4	no	—
352	C₂²×C₄	—	2	4	no	—
353	C₂²×C₄	—	2	4	no	—
354	C₂²×C₄	—	2	4	no	—
355	C₂²×C₄	—	2	2	no	—
356	C₂²×C₄	—	2	4	no	—
357	C₂²×C₄	—	2	2	no	—
358	C₂²×C₄	—	2	4	no	—
359	C₂²×C₄	—	2	4	no	—
360	C₂²×C₄	—	2	4	no	—
361	C₂²×C₄	—	2	4	no	—
362	C₂²×C₄	#41×C₆₇	1	4	yes	—
363	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
364	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
365	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
366	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
367	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
368	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
369	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
370	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
371	C₄⋊C₄	—	4	4	no	—
372	C₄⋊C₄	—	4	8	no	—
373	C₄⋊C₄	—	4	8	no	—
374	C₂²⋊C₄	—	2	16	no	—
375	C₂²⋊C₄	—	4	8	no	—
376	C₂²⋊C₄	—	4	8	no	—
377	C₂²⋊C₄	—	4	8	no	—
378	C₂²⋊C₄	—	4	8	no	—
379	C₂²⋊C₄	—	4	8	no	—
380	C₄²	—	4	4	no	—
381	C₂³	—	2	4	no	—
382	C₂³	—	2	4	no	—
383	C₂³	—	2	4	no	—
384	C₂³	—	2	4	no	—
385	C₂³	—	2	4	no	—
386	C₂³	—	2	4	no	—
387	C₂³	—	2	4	no	—
388	C₂³	—	4	8	no	—
389	C₂³	—	4	8	no	—
390	C₂³	—	4	8	no	—
391	C₂³	—	4	8	no	—
392	C₂³	—	4	8	no	—
393	C₂³	—	4	8	no	—
394	C₂³	—	4	8	no	—
395	C₂³	C₂×#41×C₆₇	1	2	yes	—
396	C₂³	H(5,𝔽₂)×C₆₇	1	2	yes	—
397	C₂³	H(5,𝔽₂)×C₆₇	1	2	yes	—
398	C₂³	H(5,𝔽₂)×C₆₇	1	4	yes	—
399	C₂³	2₋¹⁺⁴×C₆₇	1	4	yes	—
400	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
401	Q₈	—	4	16	no	—
402	Q₈	—	4	16	no	—
403	D₄	—	4	16	no	—
404	C₂×C₄	—	2	8	no	—
405	C₂×C₄	—	2	2	no	—
406	C₂×C₄	—	2	2	no	—
407	C₂×C₄	—	2	4	no	—
408	C₂×C₄	—	2	4	no	—
409	C₂×C₄	—	2	4	no	—
410	C₂×C₄	—	2	4	no	—
411	C₂×C₄	—	2	4	no	—
412	C₂×C₄	—	2	4	no	—
413	C₂×C₄	—	2	4	no	—
414	C₂×C₄	—	2	4	no	—
415	C₂×C₄	—	2	4	no	—
416	C₂×C₄	—	2	4	no	—
417	C₂×C₄	—	2	4	no	—
418	C₂×C₄	—	2	4	no	—
419	C₂×C₄	—	2	4	no	—
420	C₂×C₄	—	2	2	no	—
421	C₂×C₄	—	2	4	no	—
422	C₂×C₄	—	2	2	no	—
423	C₂×C₄	—	2	4	no	—
424	C₂×C₄	—	2	4	no	—
425	C₂×C₄	—	2	4	no	—
426	C₂×C₄	—	4	8	no	—
427	C₂×C₄	—	4	8	no	—
428	C₂×C₄	—	4	4	no	—
429	C₂×C₄	—	4	8	no	—
430	C₂×C₄	—	4	4	no	—
431	C₂×C₄	—	4	8	no	—
432	C₂×C₄	—	4	8	no	—
433	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
434	C₂²	—	4	4	no	—
435	C₂²	—	4	4	no	—
436	C₂²	—	4	4	no	—
437	C₂²	—	4	4	no	—
438	C₂²	—	4	4	no	—
439	C₂²	—	4	4	no	—
440	C₂²	—	4	4	no	—
441	C₂²	—	8	16	no	—
442	C₂²	#538×C₆₇	1	4	yes	—
443	C₂²	#542×C₆₇	1	8	yes	—
444	C₂²	H(3,𝔽₄)×C₆₇	1	4	yes	—
445	C₂²	#562×C₆₇	1	8	yes	—
446	C₂²	#563×C₆₇	1	4	yes	—
447	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
448	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	2	yes	—
449	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
450	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
451	C₂²	C₂×2₋¹⁺⁴×C₆₇	1	2	yes	—
452	C₄	—	2	4	no	—
453	C₄	—	4	2	no	—
454	C₄	—	4	2	no	—
455	C₄	—	4	4	no	—
456	C₄	—	4	4	no	—
457	C₄	—	4	4	no	—
458	C₄	—	4	4	no	—
459	C₄	—	4	4	no	—
460	C₂	—	8	4	no	—
461	C₂	C₆₇×#2832	1	8	yes	direct product factor
462	C₂	C₂×#538×C₆₇	1	1	yes	characteristic
463	C₂	C₂×#542×C₆₇	1	2	yes	—
464	C₂	C₂×H(3,𝔽₄)×C₆₇	1	1	yes	characteristic
465	C₂	C₂×#562×C₆₇	1	2	yes	—
466	C₂	C₂×#563×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
1. C₆₇×#2832	C₂	direct (N × H)
2. C₂×#400×C₆₇	C₂	semidirect
3. C₂×#397×C₆₇	C₂	semidirect
4. C₂×#395×C₆₇	C₂	semidirect
7. C₂×#393×C₆₇	C₂	semidirect
8. C₂×#393×C₆₇	C₂	semidirect
9. C₂×#391×C₆₇	C₂	semidirect
10. C₂×#388×C₆₇	C₂	semidirect
24. #400×C₆₇	C₂²	semidirect
25. #397×C₆₇	C₂²	semidirect
26. #395×C₆₇	C₂²	semidirect
29. #393×C₆₇	C₂²	semidirect
30. #393×C₆₇	C₂²	semidirect
31. #391×C₆₇	C₂²	semidirect
32. #388×C₆₇	C₂²	semidirect
226. C₂×#2832	C₆₇	direct (N × H)
228. C₂×C₆₇	#2832	direct (N × H)
234. #2832	C₂×C₆₇	direct (N × H)
235. C₂×#400	C₂×C₆₇	semidirect
236. C₂×#397	C₂×C₆₇	semidirect
237. C₂×#395	C₂×C₆₇	semidirect
240. C₂×#393	C₂×C₆₇	semidirect
241. C₂×#393	C₂×C₆₇	semidirect
242. C₂×#391	C₂×C₆₇	semidirect
243. C₂×#388	C₂×C₆₇	semidirect
244. C₆₇	C₂×#2832	direct (N × H)
258. #400	C₂²×C₆₇	semidirect
259. #397	C₂²×C₆₇	semidirect
260. #395	C₂²×C₆₇	semidirect
263. #393	C₂²×C₆₇	semidirect
264. #393	C₂²×C₆₇	semidirect
265. #391	C₂²×C₆₇	semidirect
266. #388	C₂²×C₆₇	semidirect
461. C₂	C₆₇×#2832	direct (N × H)

Maximal subgroups

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

subgroup	index
C₆₇×#2832	2
C₂×#400×C₆₇	2
C₂×#397×C₆₇	2
C₂×#395×C₆₇	2
C₂×#394×C₆₇	2
C₂×#393×C₆₇	2
C₂×#391×C₆₇	2
C₂×#388×C₆₇	2
C₂×#2832	67

Subgroups — C2×C67×#2832 #30715916

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#2832 #30715916