Subgroups — C₂×C₆₇×#2637 #30713852

Order 17152.

2482 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#126×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2637	C₂	1	16	yes	maximal, direct product factor
4	C₂×#398×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#398×C₆₇	C₂	1	2	yes	maximal, complemented
6	C₂×#397×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂×#396×C₆₇	C₂	1	2	yes	maximal
8	C₂×#395×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#393×C₆₇	C₂	1	1	yes	maximal, characteristic
10	C₂×#388×C₆₇	C₂	1	2	yes	maximal
11	C₂×#387×C₆₇	C₂	1	2	yes	maximal, complemented
12	C₂×#382×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂³×Q₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×#126×C₆₇	—	2	8	no	—
15	C₂×#126×C₆₇	C₂²	1	8	yes	—
16	C₂×C₄×Q₈×C₆₇	—	2	8	no	—
17	C₂×C₄×Q₈×C₆₇	C₂²	1	8	yes	complemented
18	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
19	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
20	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
21	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
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×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
26	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
27	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
28	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
29	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
30	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
31	C₂²×C₄²×C₆₇	C₂²	1	2	yes	—
32	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
33	#398×C₆₇	C₂²	1	16	yes	complemented
34	#398×C₆₇	C₂²	1	8	yes	complemented
35	#397×C₆₇	C₂²	1	8	yes	—
36	#396×C₆₇	C₂²	1	16	yes	—
37	#395×C₆₇	C₂²	1	8	yes	complemented
38	#393×C₆₇	C₂²	1	8	yes	—
39	#388×C₆₇	C₂²	1	16	yes	—
40	#387×C₆₇	C₂²	1	16	yes	complemented
41	#382×C₆₇	C₂²	1	8	yes	complemented
42	C₂×#95×C₆₇	C₂²	1	2	yes	—
43	C₂×#95×C₆₇	C₂²	1	1	yes	characteristic
44	C₂×#95×C₆₇	C₂²	1	1	yes	characteristic
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	1	yes	characteristic
50	C₂²×Q₈×C₆₇	—	2	4	no	—
51	C₂²×Q₈×C₆₇	—	2	8	no	—
52	C₂²×Q₈×C₆₇	C₂³	1	4	yes	—
53	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
54	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
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	2	yes	—
59	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
60	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
61	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
62	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
63	#126×C₆₇	—	2	32	no	—
64	C₄×Q₈×C₆₇	—	2	32	no	—
65	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
66	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
67	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
68	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
69	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
70	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
71	C₂×C₄⋊C₄×C₆₇	—	4	4	no	—
72	C₂×C₄⋊C₄×C₆₇	—	4	4	no	—
73	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
74	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
75	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
76	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
77	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
78	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
79	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
80	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
81	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
82	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
83	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
84	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
85	C₂×C₂²⋊C₄×C₆₇	—	4	4	no	—
86	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
87	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
88	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
89	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
90	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
91	C₂×C₄²×C₆₇	—	2	8	no	—
92	C₂×C₄²×C₆₇	D₄	1	8	yes	—
93	C₂×C₄²×C₆₇	C₂³	1	8	yes	—
94	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
95	#95×C₆₇	C₂³	1	8	yes	—
96	#95×C₆₇	C₂³	1	4	yes	—
97	#95×C₆₇	C₂³	1	4	yes	—
98	#95×C₆₇	C₂³	1	8	yes	—
99	#95×C₆₇	C₂³	1	8	yes	—
100	#95×C₆₇	C₂³	1	8	yes	—
101	#95×C₆₇	C₂³	1	8	yes	—
102	C₂⁴×C₆₇	—	2	2	no	—
103	C₂⁴×C₆₇	—	2	4	no	—
104	C₂⁴×C₆₇	—	2	2	no	—
105	C₂⁴×C₆₇	—	2	4	no	—
106	C₂⁴×C₆₇	—	2	2	no	—
107	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
108	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
109	C₂×Q₈×C₆₇	—	2	32	no	—
110	C₂×Q₈×C₆₇	—	2	8	no	—
111	C₂×Q₈×C₆₇	—	2	16	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	2	no	—
116	C₂²×C₄×C₆₇	—	2	4	no	—
117	C₂²×C₄×C₆₇	—	2	2	no	—
118	C₂²×C₄×C₆₇	—	2	4	no	—
119	C₂²×C₄×C₆₇	—	2	2	no	—
120	C₂²×C₄×C₆₇	—	2	2	no	—
121	C₂²×C₄×C₆₇	—	2	2	no	—
122	C₂²×C₄×C₆₇	—	2	4	no	—
123	C₂²×C₄×C₆₇	—	2	4	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	2	no	—
128	C₂²×C₄×C₆₇	—	2	4	no	—
129	C₂²×C₄×C₆₇	—	2	4	no	—
130	C₂²×C₄×C₆₇	—	2	4	no	—
131	C₂²×C₄×C₆₇	—	2	2	no	—
132	C₂²×C₄×C₆₇	—	2	4	no	—
133	C₂²×C₄×C₆₇	—	2	4	no	—
134	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
135	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
136	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
137	C₂²×C₄×C₆₇	#41	1	8	yes	—
138	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
139	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
140	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
141	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
142	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
143	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
144	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
145	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
146	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
147	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
148	C₄⋊C₄×C₆₇	—	2	16	no	—
149	C₄⋊C₄×C₆₇	—	2	8	no	—
150	C₄⋊C₄×C₆₇	—	2	8	no	—
151	C₄⋊C₄×C₆₇	—	2	8	no	—
152	C₄⋊C₄×C₆₇	—	2	8	no	—
153	C₄⋊C₄×C₆₇	—	2	16	no	—
154	C₄⋊C₄×C₆₇	—	4	8	no	—
155	C₄⋊C₄×C₆₇	—	4	8	no	—
156	C₂²⋊C₄×C₆₇	—	2	8	no	—
157	C₂²⋊C₄×C₆₇	—	2	8	no	—
158	C₂²⋊C₄×C₆₇	—	2	8	no	—
159	C₂²⋊C₄×C₆₇	—	2	16	no	—
160	C₂²⋊C₄×C₆₇	—	4	8	no	—
161	C₄²×C₆₇	—	2	16	no	—
162	C₄²×C₆₇	C₂×D₄	1	16	yes	—
163	C₂³×C₆₇	—	2	2	no	—
164	C₂³×C₆₇	—	2	4	no	—
165	C₂³×C₆₇	—	2	2	no	—
166	C₂³×C₆₇	—	2	4	no	—
167	C₂³×C₆₇	—	2	2	no	—
168	C₂³×C₆₇	—	4	4	no	—
169	C₂³×C₆₇	—	4	4	no	—
170	C₂³×C₆₇	—	4	4	no	—
171	C₂³×C₆₇	—	4	8	no	—
172	C₂³×C₆₇	—	4	8	no	—
173	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
174	C₂³×C₆₇	C₂×#41	1	4	yes	—
175	C₂³×C₆₇	H(5,𝔽₂)	1	2	yes	—
176	C₂³×C₆₇	2₋¹⁺⁴	1	4	yes	—
177	C₂³×C₆₇	2₋¹⁺⁴	1	2	yes	—
178	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
179	Q₈×C₆₇	—	2	32	no	—
180	C₂×C₄×C₆₇	—	2	8	no	—
181	C₂×C₄×C₆₇	—	2	16	no	—
182	C₂×C₄×C₆₇	—	2	8	no	—
183	C₂×C₄×C₆₇	—	2	8	no	—
184	C₂×C₄×C₆₇	—	2	4	no	—
185	C₂×C₄×C₆₇	—	2	4	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	2	no	—
190	C₂×C₄×C₆₇	—	2	4	no	—
191	C₂×C₄×C₆₇	—	2	2	no	—
192	C₂×C₄×C₆₇	—	2	2	no	—
193	C₂×C₄×C₆₇	—	2	2	no	—
194	C₂×C₄×C₆₇	—	2	4	no	—
195	C₂×C₄×C₆₇	—	2	4	no	—
196	C₂×C₄×C₆₇	—	2	4	no	—
197	C₂×C₄×C₆₇	—	2	4	no	—
198	C₂×C₄×C₆₇	—	2	4	no	—
199	C₂×C₄×C₆₇	—	2	2	no	—
200	C₂×C₄×C₆₇	—	2	4	no	—
201	C₂×C₄×C₆₇	—	2	4	no	—
202	C₂×C₄×C₆₇	—	2	4	no	—
203	C₂×C₄×C₆₇	—	2	2	no	—
204	C₂×C₄×C₆₇	—	2	4	no	—
205	C₂×C₄×C₆₇	—	2	4	no	—
206	C₂×C₄×C₆₇	—	4	4	no	—
207	C₂×C₄×C₆₇	—	4	4	no	—
208	C₂×C₄×C₆₇	—	4	8	no	—
209	C₂×C₄×C₆₇	—	4	4	no	—
210	C₂×C₄×C₆₇	—	4	8	no	—
211	C₂×C₄×C₆₇	—	4	8	no	—
212	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
213	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
214	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
215	C₂×C₄×C₆₇	C₂×#41	1	8	yes	—
216	C₂²×C₆₇	—	4	2	no	—
217	C₂²×C₆₇	—	4	2	no	—
218	C₂²×C₆₇	—	4	2	no	—
219	C₂²×C₆₇	—	4	4	no	—
220	C₂²×C₆₇	—	4	4	no	—
221	C₂²×C₆₇	—	8	8	no	—
222	C₂²×C₆₇	#525	1	4	yes	—
223	C₂²×C₆₇	#536	1	4	yes	—
224	C₂²×C₆₇	#537	1	4	yes	—
225	C₂²×C₆₇	#541	1	8	yes	—
226	C₂²×C₆₇	#542	1	8	yes	—
227	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
228	C₂²×C₆₇	C₂²×#41	1	2	yes	—
229	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
230	C₂²×C₆₇	C₂×2₋¹⁺⁴	1	2	yes	—
231	C₂²×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
232	C₄×C₆₇	—	2	4	no	—
233	C₄×C₆₇	—	2	8	no	—
234	C₄×C₆₇	—	2	4	no	—
235	C₄×C₆₇	—	2	4	no	—
236	C₄×C₆₇	—	4	2	no	—
237	C₄×C₆₇	—	4	2	no	—
238	C₄×C₆₇	—	4	4	no	—
239	C₄×C₆₇	—	4	2	no	—
240	C₄×C₆₇	—	4	4	no	—
241	C₄×C₆₇	—	4	4	no	—
242	C₂×#2637	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
243	C₂×C₆₇	—	8	2	no	—
244	C₂×C₆₇	#2637	1	8	yes	direct product factor
245	C₂×C₆₇	C₂×#525	1	1	yes	characteristic
246	C₂×C₆₇	C₂×#536	1	1	yes	characteristic
247	C₂×C₆₇	C₂×#537	1	1	yes	characteristic
248	C₂×C₆₇	C₂×#541	1	2	yes	—
249	C₂×C₆₇	C₂×#542	1	2	yes	—
250	C₂²×#126	C₂×C₆₇	1	1	yes	characteristic
251	C₂²×C₄×Q₈	C₂×C₆₇	1	1	yes	characteristic, complemented
252	#2637	C₂×C₆₇	1	16	yes	direct product factor
253	C₂×#398	C₂×C₆₇	1	1	yes	characteristic, complemented
254	C₂×#398	C₂×C₆₇	1	2	yes	complemented
255	C₂×#397	C₂×C₆₇	1	1	yes	characteristic
256	C₂×#396	C₂×C₆₇	1	2	yes	—
257	C₂×#395	C₂×C₆₇	1	1	yes	characteristic, complemented
258	C₂×#393	C₂×C₆₇	1	1	yes	characteristic
259	C₂×#388	C₂×C₆₇	1	2	yes	—
260	C₂×#387	C₂×C₆₇	1	2	yes	complemented
261	C₂×#382	C₂×C₆₇	1	1	yes	characteristic, complemented
262	C₆₇	C₂×#2637	1	1	yes	characteristic, direct product factor, Sylow
263	C₂³×Q₈	C₂²×C₆₇	1	1	yes	characteristic
264	C₂×#126	—	2	8	no	—
265	C₂×#126	C₂²×C₆₇	1	8	yes	—
266	C₂×C₄×Q₈	—	2	8	no	—
267	C₂×C₄×Q₈	C₂²×C₆₇	1	8	yes	complemented
268	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
269	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
270	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
271	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
272	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
273	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
274	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
275	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
276	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
277	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
278	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
279	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
280	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
281	C₂²×C₄²	C₂²×C₆₇	1	2	yes	—
282	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
283	#398	C₂²×C₆₇	1	8	yes	complemented
284	#398	C₂²×C₆₇	1	16	yes	complemented
285	#397	C₂²×C₆₇	1	8	yes	—
286	#396	C₂²×C₆₇	1	16	yes	—
287	#395	C₂²×C₆₇	1	8	yes	complemented
288	#393	C₂²×C₆₇	1	8	yes	—
289	#388	C₂²×C₆₇	1	16	yes	—
290	#387	C₂²×C₆₇	1	16	yes	complemented
291	#382	C₂²×C₆₇	1	8	yes	complemented
292	C₂×#95	C₂²×C₆₇	1	1	yes	characteristic
293	C₂×#95	C₂²×C₆₇	1	2	yes	—
294	C₂×#95	C₂²×C₆₇	1	1	yes	characteristic
295	C₂×#95	C₂²×C₆₇	1	2	yes	—
296	C₂×#95	C₂²×C₆₇	1	2	yes	—
297	C₂×#95	C₂²×C₆₇	1	2	yes	—
298	C₂×#95	C₂²×C₆₇	1	2	yes	—
299	C₂⁵	C₂³×C₆₇	1	1	yes	characteristic
300	C₂²×Q₈	—	2	4	no	—
301	C₂²×Q₈	—	2	8	no	—
302	C₂²×Q₈	C₂³×C₆₇	1	4	yes	—
303	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
304	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
305	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
306	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
307	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
308	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
309	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
310	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
311	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
312	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
313	#126	—	2	32	no	—
314	C₄×Q₈	—	2	32	no	—
315	C₂×C₄⋊C₄	—	2	8	no	—
316	C₂×C₄⋊C₄	—	2	4	no	—
317	C₂×C₄⋊C₄	—	2	4	no	—
318	C₂×C₄⋊C₄	—	2	4	no	—
319	C₂×C₄⋊C₄	—	2	4	no	—
320	C₂×C₄⋊C₄	—	2	8	no	—
321	C₂×C₄⋊C₄	—	4	4	no	—
322	C₂×C₄⋊C₄	—	4	4	no	—
323	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
324	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
325	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
326	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
327	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
328	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
329	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
330	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
331	C₂×C₂²⋊C₄	—	2	4	no	—
332	C₂×C₂²⋊C₄	—	2	4	no	—
333	C₂×C₂²⋊C₄	—	2	4	no	—
334	C₂×C₂²⋊C₄	—	2	8	no	—
335	C₂×C₂²⋊C₄	—	4	4	no	—
336	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
337	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
338	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
339	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
340	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
341	C₂×C₄²	—	2	8	no	—
342	C₂×C₄²	D₄×C₆₇	1	8	yes	—
343	C₂×C₄²	C₂³×C₆₇	1	8	yes	—
344	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
345	#95	C₂³×C₆₇	1	8	yes	—
346	#95	C₂³×C₆₇	1	4	yes	—
347	#95	C₂³×C₆₇	1	4	yes	—
348	#95	C₂³×C₆₇	1	8	yes	—
349	#95	C₂³×C₆₇	1	8	yes	—
350	#95	C₂³×C₆₇	1	8	yes	—
351	#95	C₂³×C₆₇	1	8	yes	—
352	C₂⁴	—	2	2	no	—
353	C₂⁴	—	2	4	no	—
354	C₂⁴	—	2	2	no	—
355	C₂⁴	—	2	4	no	—
356	C₂⁴	—	2	2	no	—
357	C₂⁴	C₂⁴×C₆₇	1	2	yes	—
358	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
359	C₂×Q₈	—	2	32	no	—
360	C₂×Q₈	—	2	8	no	—
361	C₂×Q₈	—	2	16	no	—
362	C₂²×C₄	—	2	4	no	—
363	C₂²×C₄	—	2	4	no	—
364	C₂²×C₄	—	2	4	no	—
365	C₂²×C₄	—	2	2	no	—
366	C₂²×C₄	—	2	4	no	—
367	C₂²×C₄	—	2	2	no	—
368	C₂²×C₄	—	2	4	no	—
369	C₂²×C₄	—	2	2	no	—
370	C₂²×C₄	—	2	2	no	—
371	C₂²×C₄	—	2	2	no	—
372	C₂²×C₄	—	2	4	no	—
373	C₂²×C₄	—	2	4	no	—
374	C₂²×C₄	—	2	4	no	—
375	C₂²×C₄	—	2	4	no	—
376	C₂²×C₄	—	2	4	no	—
377	C₂²×C₄	—	2	2	no	—
378	C₂²×C₄	—	2	4	no	—
379	C₂²×C₄	—	2	4	no	—
380	C₂²×C₄	—	2	2	no	—
381	C₂²×C₄	—	2	4	no	—
382	C₂²×C₄	—	2	4	no	—
383	C₂²×C₄	—	2	4	no	—
384	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
385	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
386	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
387	C₂²×C₄	#41×C₆₇	1	8	yes	—
388	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
389	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
390	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
391	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
392	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
393	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
394	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
395	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
396	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
397	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
398	C₄⋊C₄	—	2	16	no	—
399	C₄⋊C₄	—	2	8	no	—
400	C₄⋊C₄	—	2	8	no	—
401	C₄⋊C₄	—	2	8	no	—
402	C₄⋊C₄	—	2	8	no	—
403	C₄⋊C₄	—	2	16	no	—
404	C₄⋊C₄	—	4	8	no	—
405	C₄⋊C₄	—	4	8	no	—
406	C₂²⋊C₄	—	2	8	no	—
407	C₂²⋊C₄	—	2	8	no	—
408	C₂²⋊C₄	—	2	8	no	—
409	C₂²⋊C₄	—	2	16	no	—
410	C₂²⋊C₄	—	4	8	no	—
411	C₄²	—	2	16	no	—
412	C₄²	C₂×D₄×C₆₇	1	16	yes	—
413	C₂³	—	2	2	no	—
414	C₂³	—	2	4	no	—
415	C₂³	—	2	2	no	—
416	C₂³	—	2	4	no	—
417	C₂³	—	2	2	no	—
418	C₂³	—	4	4	no	—
419	C₂³	—	4	4	no	—
420	C₂³	—	4	4	no	—
421	C₂³	—	4	8	no	—
422	C₂³	—	4	8	no	—
423	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
424	C₂³	C₂×#41×C₆₇	1	4	yes	—
425	C₂³	H(5,𝔽₂)×C₆₇	1	2	yes	—
426	C₂³	2₋¹⁺⁴×C₆₇	1	4	yes	—
427	C₂³	2₋¹⁺⁴×C₆₇	1	2	yes	—
428	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
429	Q₈	—	2	32	no	—
430	C₂×C₄	—	2	8	no	—
431	C₂×C₄	—	2	16	no	—
432	C₂×C₄	—	2	8	no	—
433	C₂×C₄	—	2	8	no	—
434	C₂×C₄	—	2	4	no	—
435	C₂×C₄	—	2	4	no	—
436	C₂×C₄	—	2	4	no	—
437	C₂×C₄	—	2	2	no	—
438	C₂×C₄	—	2	4	no	—
439	C₂×C₄	—	2	2	no	—
440	C₂×C₄	—	2	4	no	—
441	C₂×C₄	—	2	2	no	—
442	C₂×C₄	—	2	2	no	—
443	C₂×C₄	—	2	2	no	—
444	C₂×C₄	—	2	4	no	—
445	C₂×C₄	—	2	4	no	—
446	C₂×C₄	—	2	4	no	—
447	C₂×C₄	—	2	4	no	—
448	C₂×C₄	—	2	4	no	—
449	C₂×C₄	—	2	2	no	—
450	C₂×C₄	—	2	4	no	—
451	C₂×C₄	—	2	4	no	—
452	C₂×C₄	—	2	4	no	—
453	C₂×C₄	—	2	2	no	—
454	C₂×C₄	—	2	4	no	—
455	C₂×C₄	—	2	4	no	—
456	C₂×C₄	—	4	4	no	—
457	C₂×C₄	—	4	4	no	—
458	C₂×C₄	—	4	8	no	—
459	C₂×C₄	—	4	4	no	—
460	C₂×C₄	—	4	8	no	—
461	C₂×C₄	—	4	8	no	—
462	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
463	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
464	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
465	C₂×C₄	C₂×#41×C₆₇	1	8	yes	—
466	C₂²	—	4	2	no	—
467	C₂²	—	4	2	no	—
468	C₂²	—	4	2	no	—
469	C₂²	—	4	4	no	—
470	C₂²	—	4	4	no	—
471	C₂²	—	8	8	no	—
472	C₂²	#525×C₆₇	1	4	yes	—
473	C₂²	#536×C₆₇	1	4	yes	—
474	C₂²	#537×C₆₇	1	4	yes	—
475	C₂²	#541×C₆₇	1	8	yes	—
476	C₂²	#542×C₆₇	1	8	yes	—
477	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
478	C₂²	C₂²×#41×C₆₇	1	2	yes	—
479	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
480	C₂²	C₂×2₋¹⁺⁴×C₆₇	1	2	yes	—
481	C₂²	C₂×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
482	C₄	—	2	4	no	—
483	C₄	—	2	8	no	—
484	C₄	—	2	4	no	—
485	C₄	—	2	4	no	—
486	C₄	—	4	2	no	—
487	C₄	—	4	2	no	—
488	C₄	—	4	4	no	—
489	C₄	—	4	2	no	—
490	C₄	—	4	4	no	—
491	C₄	—	4	4	no	—
492	C₂	—	8	2	no	—
493	C₂	C₆₇×#2637	1	8	yes	direct product factor
494	C₂	C₂×#525×C₆₇	1	1	yes	characteristic
495	C₂	C₂×#536×C₆₇	1	1	yes	characteristic
496	C₂	C₂×#537×C₆₇	1	1	yes	characteristic
497	C₂	C₂×#541×C₆₇	1	2	yes	—
498	C₂	C₂×#542×C₆₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₆₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₆₇
prime radical	C₂⁵×C₆₇	C₂³
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×C₄×Q₈×C₆₇	C₂	semidirect
3. C₆₇×#2637	C₂	direct (N × H)
4. C₂×#398×C₆₇	C₂	semidirect
5. C₂×#398×C₆₇	C₂	semidirect
8. C₂×#395×C₆₇	C₂	semidirect
11. C₂×#387×C₆₇	C₂	semidirect
12. C₂×#382×C₆₇	C₂	semidirect
17. C₂×C₄×Q₈×C₆₇	C₂²	semidirect
33. #398×C₆₇	C₂²	semidirect
34. #398×C₆₇	C₂²	semidirect
37. #395×C₆₇	C₂²	semidirect
40. #387×C₆₇	C₂²	semidirect
41. #382×C₆₇	C₂²	semidirect
242. C₂×#2637	C₆₇	direct (N × H)
244. C₂×C₆₇	#2637	direct (N × H)
251. C₂²×C₄×Q₈	C₂×C₆₇	semidirect
252. #2637	C₂×C₆₇	direct (N × H)
253. C₂×#398	C₂×C₆₇	semidirect
254. C₂×#398	C₂×C₆₇	semidirect
257. C₂×#395	C₂×C₆₇	semidirect
260. C₂×#387	C₂×C₆₇	semidirect
261. C₂×#382	C₂×C₆₇	semidirect
262. C₆₇	C₂×#2637	direct (N × H)
267. C₂×C₄×Q₈	C₂²×C₆₇	semidirect
283. #398	C₂²×C₆₇	semidirect
284. #398	C₂²×C₆₇	semidirect
287. #395	C₂²×C₆₇	semidirect
290. #387	C₂²×C₆₇	semidirect
291. #382	C₂²×C₆₇	semidirect
493. C₂	C₆₇×#2637	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×#126×C₆₇	2
C₂²×C₄×Q₈×C₆₇	2
C₆₇×#2637	2
C₂×#398×C₆₇	2
C₂×#397×C₆₇	2
C₂×#396×C₆₇	2
C₂×#395×C₆₇	2
C₂×#393×C₆₇	2
C₂×#388×C₆₇	2
C₂×#387×C₆₇	2
C₂×#382×C₆₇	2
C₂×#2637	67

Subgroups — C2×C67×#2637 #30713852

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#2637 #30713852