Subgroups — C₂×C₆₇×#3109 #30739792

Order 17152.

2830 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#532×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₄×D₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#3109	C₂	1	16	yes	maximal, direct product factor
4	C₂×#481×C₆₇	C₂	1	2	yes	maximal, complemented
5	C₂×#476×C₆₇	C₂	1	2	yes	maximal, complemented
6	C₂×#459×C₆₇	C₂	1	2	yes	maximal, complemented
7	C₂×#447×C₆₇	C₂	1	2	yes	maximal, complemented
8	C₂×#442×C₆₇	C₂	1	4	yes	maximal, complemented
9	C₂×#432×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#41×C₆₇	—	2	1	no	—
11	C₂³×D₄×C₆₇	C₂²	1	1	yes	characteristic
12	C₂⁴×C₄×C₆₇	—	2	1	no	—
13	C₂²×D₈×C₆₇	C₂²	1	4	yes	—
14	#532×C₆₇	—	2	4	no	—
15	#532×C₆₇	C₂²	1	8	yes	complemented
16	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
17	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic, complemented
18	C₂×#124×C₆₇	—	2	1	no	—
19	C₂×#121×C₆₇	—	2	1	no	—
20	C₂×#121×C₆₇	C₂²	1	2	yes	—
21	C₂×C₄×D₄×C₆₇	—	2	1	no	—
22	C₂×C₄×D₄×C₆₇	—	2	1	no	—
23	C₂×C₄×D₄×C₆₇	—	2	4	no	—
24	C₂×C₄×D₄×C₆₇	—	2	1	no	—
25	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
26	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
27	C₂×C₄×D₄×C₆₇	C₂²	1	8	yes	complemented
28	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	complemented
29	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
30	C₂²×C₂²⋊C₄×C₆₇	—	2	1	no	—
31	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
32	#481×C₆₇	C₂²	1	16	yes	complemented
33	#476×C₆₇	C₂²	1	16	yes	complemented
34	#459×C₆₇	C₂²	1	16	yes	complemented
35	#447×C₆₇	C₂²	1	16	yes	complemented
36	#442×C₆₇	C₂²	1	32	yes	complemented
37	#432×C₆₇	C₂²	1	8	yes	complemented
38	C₂×#106×C₆₇	C₂²	1	4	yes	complemented
39	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	complemented
40	C₂×#102×C₆₇	C₂²	1	4	yes	—
41	C₂×#102×C₆₇	C₂²	1	4	yes	complemented
42	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
43	C₂×#97×C₆₇	C₂²	1	2	yes	complemented
44	C₂⁵×C₆₇	—	2	1	no	—
45	C₂×#41×C₆₇	—	2	8	no	—
46	C₂×#41×C₆₇	—	4	8	no	—
47	C₂²×Q₈×C₆₇	—	2	1	no	—
48	C₂²×D₄×C₆₇	—	2	8	no	—
49	C₂²×D₄×C₆₇	—	2	2	no	—
50	C₂²×D₄×C₆₇	—	2	2	no	—
51	C₂²×D₄×C₆₇	—	2	1	no	—
52	C₂²×D₄×C₆₇	—	2	1	no	—
53	C₂²×D₄×C₆₇	—	2	1	no	—
54	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
55	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
56	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
57	C₂³×C₄×C₆₇	—	2	1	no	—
58	C₂³×C₄×C₆₇	—	2	2	no	—
59	C₂³×C₄×C₆₇	—	2	2	no	—
60	C₂³×C₄×C₆₇	—	2	8	no	—
61	C₂³×C₄×C₆₇	—	2	2	no	—
62	C₂³×C₄×C₆₇	—	4	8	no	—
63	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
64	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
65	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
66	C₂×D₈×C₆₇	—	2	16	no	—
67	C₂×D₈×C₆₇	C₂³	1	16	yes	—
68	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
69	C₄⋊Q₈×C₆₇	—	2	2	no	—
70	C₄⋊Q₈×C₆₇	C₂³	1	4	yes	complemented
71	#127×C₆₇	—	2	2	no	—
72	#127×C₆₇	C₂³	1	4	yes	complemented
73	#124×C₆₇	—	2	8	no	—
74	#121×C₆₇	—	2	8	no	—
75	#121×C₆₇	—	2	4	no	—
76	#121×C₆₇	C₂³	1	8	yes	—
77	C₄×D₄×C₆₇	—	2	4	no	—
78	C₄×D₄×C₆₇	—	2	4	no	—
79	C₄×D₄×C₆₇	—	2	8	no	—
80	C₄×D₄×C₆₇	—	2	8	no	—
81	C₄×D₄×C₆₇	—	2	8	no	—
82	C₄×D₄×C₆₇	—	2	4	no	—
83	C₄×D₄×C₆₇	C₂³	1	8	yes	—
84	C₄×D₄×C₆₇	C₂³	1	8	yes	—
85	C₄×D₄×C₆₇	C₂³	1	8	yes	complemented
86	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
87	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
88	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
89	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
90	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
91	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
92	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
93	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
94	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
95	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
96	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
97	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
98	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
99	C₂×C₄²×C₆₇	D₄	1	4	yes	—
100	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
101	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
102	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
103	#106×C₆₇	C₂³	1	16	yes	complemented
104	C₄⋊C₈×C₆₇	C₂³	1	8	yes	complemented
105	#102×C₆₇	C₂³	1	16	yes	—
106	#102×C₆₇	C₂³	1	16	yes	complemented
107	C₂²⋊C₈×C₆₇	C₂³	1	8	yes	—
108	#97×C₆₇	C₂³	1	8	yes	complemented
109	C₂⁴×C₆₇	—	2	8	no	—
110	C₂⁴×C₆₇	—	2	4	no	—
111	C₂⁴×C₆₇	—	2	2	no	—
112	C₂⁴×C₆₇	—	2	2	no	—
113	C₂⁴×C₆₇	—	4	4	no	—
114	C₂⁴×C₆₇	—	4	4	no	—
115	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
116	#41×C₆₇	—	4	32	no	—
117	C₂×Q₈×C₆₇	—	2	4	no	—
118	C₂×Q₈×C₆₇	—	4	4	no	—
119	C₂×D₄×C₆₇	—	2	8	no	—
120	C₂×D₄×C₆₇	—	2	16	no	—
121	C₂×D₄×C₆₇	—	2	16	no	—
122	C₂×D₄×C₆₇	—	2	8	no	—
123	C₂×D₄×C₆₇	—	2	4	no	—
124	C₂×D₄×C₆₇	—	2	2	no	—
125	C₂×D₄×C₆₇	—	2	2	no	—
126	C₂×D₄×C₆₇	—	2	4	no	—
127	C₂×D₄×C₆₇	—	2	4	no	—
128	C₂×D₄×C₆₇	—	2	4	no	—
129	C₂×D₄×C₆₇	—	2	8	no	—
130	C₂×D₄×C₆₇	—	4	4	no	—
131	C₂×D₄×C₆₇	—	4	8	no	—
132	C₂×D₄×C₆₇	—	4	4	no	—
133	C₂×D₄×C₆₇	#41	1	8	yes	complemented
134	C₂×D₄×C₆₇	C₂⁴	1	4	yes	—
135	C₂×D₄×C₆₇	C₂⁴	1	4	yes	—
136	C₂²×C₄×C₆₇	—	2	4	no	—
137	C₂²×C₄×C₆₇	—	2	4	no	—
138	C₂²×C₄×C₆₇	—	2	4	no	—
139	C₂²×C₄×C₆₇	—	2	4	no	—
140	C₂²×C₄×C₆₇	—	2	2	no	—
141	C₂²×C₄×C₆₇	—	2	2	no	—
142	C₂²×C₄×C₆₇	—	2	8	no	—
143	C₂²×C₄×C₆₇	—	2	8	no	—
144	C₂²×C₄×C₆₇	—	2	8	no	—
145	C₂²×C₄×C₆₇	—	4	8	no	—
146	C₂²×C₄×C₆₇	—	4	32	no	—
147	C₂²×C₄×C₆₇	—	4	8	no	—
148	C₂²×C₄×C₆₇	—	4	4	no	—
149	C₂²×C₄×C₆₇	—	4	8	no	—
150	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
151	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
152	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
153	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
154	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
155	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
156	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
157	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
158	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
159	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
160	D₈×C₆₇	—	2	32	no	—
161	C₂×C₈×C₆₇	—	2	8	no	—
162	C₂×C₈×C₆₇	C₂⁴	1	8	yes	—
163	C₄⋊C₄×C₆₇	—	2	4	no	—
164	C₄⋊C₄×C₆₇	—	2	2	no	—
165	C₄⋊C₄×C₆₇	—	2	4	no	—
166	C₄⋊C₄×C₆₇	—	2	2	no	—
167	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
168	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
169	C₂²⋊C₄×C₆₇	—	2	8	no	—
170	C₂²⋊C₄×C₆₇	—	2	8	no	—
171	C₂²⋊C₄×C₆₇	—	2	8	no	—
172	C₂²⋊C₄×C₆₇	—	2	8	no	—
173	C₄²×C₆₇	C₂×D₄	1	4	yes	—
174	C₄²×C₆₇	C₂×D₄	1	2	yes	—
175	C₄²×C₆₇	C₂×D₄	1	4	yes	—
176	C₄²×C₆₇	C₂×D₄	1	2	yes	—
177	C₄²×C₆₇	C₂⁴	1	2	yes	—
178	C₄²×C₆₇	C₂⁴	1	2	yes	—
179	C₂³×C₆₇	—	2	2	no	—
180	C₂³×C₆₇	—	2	2	no	—
181	C₂³×C₆₇	—	2	16	no	—
182	C₂³×C₆₇	—	2	8	no	—
183	C₂³×C₆₇	—	2	4	no	—
184	C₂³×C₆₇	—	2	4	no	—
185	C₂³×C₆₇	—	4	8	no	—
186	C₂³×C₆₇	—	4	32	no	—
187	C₂³×C₆₇	—	4	4	no	—
188	C₂³×C₆₇	—	4	8	no	—
189	C₂³×C₆₇	—	4	4	no	—
190	C₂³×C₆₇	—	4	4	no	—
191	C₂³×C₆₇	—	4	4	no	—
192	C₂³×C₆₇	—	4	4	no	—
193	C₂³×C₆₇	AGL(1,ℤ/8)	1	4	yes	—
194	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
195	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
196	Q₈×C₆₇	—	4	8	no	—
197	D₄×C₆₇	—	2	16	no	—
198	D₄×C₆₇	—	2	8	no	—
199	D₄×C₆₇	—	2	4	no	—
200	D₄×C₆₇	—	2	8	no	—
201	D₄×C₆₇	—	4	8	no	—
202	D₄×C₆₇	—	4	16	no	—
203	D₄×C₆₇	—	4	4	no	—
204	D₄×C₆₇	—	4	8	no	—
205	D₄×C₆₇	C₂×#41	1	8	yes	complemented
206	C₂×C₄×C₆₇	—	2	2	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	8	no	—
211	C₂×C₄×C₆₇	—	2	4	no	—
212	C₂×C₄×C₆₇	—	2	8	no	—
213	C₂×C₄×C₆₇	—	2	4	no	—
214	C₂×C₄×C₆₇	—	2	4	no	—
215	C₂×C₄×C₆₇	—	2	4	no	—
216	C₂×C₄×C₆₇	—	4	16	no	—
217	C₂×C₄×C₆₇	—	4	16	no	—
218	C₂×C₄×C₆₇	—	4	8	no	—
219	C₂×C₄×C₆₇	—	4	4	no	—
220	C₂×C₄×C₆₇	—	4	4	no	—
221	C₂×C₄×C₆₇	—	4	16	no	—
222	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	4	yes	—
223	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
224	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
225	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
226	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
227	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
228	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
229	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
230	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
231	C₈×C₆₇	—	2	8	no	—
232	C₂²×C₆₇	—	2	8	no	—
233	C₂²×C₆₇	—	2	2	no	—
234	C₂²×C₆₇	—	2	2	no	—
235	C₂²×C₆₇	—	2	8	no	—
236	C₂²×C₆₇	—	4	32	no	—
237	C₂²×C₆₇	—	4	16	no	—
238	C₂²×C₆₇	—	4	4	no	—
239	C₂²×C₆₇	—	4	4	no	—
240	C₂²×C₆₇	—	4	4	no	—
241	C₂²×C₆₇	—	4	4	no	—
242	C₂²×C₆₇	—	4	4	no	—
243	C₂²×C₆₇	—	8	8	no	—
244	C₂²×C₆₇	#525	1	2	yes	—
245	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	4	yes	—
246	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
247	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
248	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
249	C₄×C₆₇	—	2	2	no	—
250	C₄×C₆₇	—	2	2	no	—
251	C₄×C₆₇	—	2	2	no	—
252	C₄×C₆₇	—	2	2	no	—
253	C₄×C₆₇	—	4	4	no	—
254	C₄×C₆₇	—	4	4	no	—
255	C₄×C₆₇	#525	1	2	yes	—
256	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	4	yes	—
257	C₄×C₆₇	C₂²×#41	1	2	yes	—
258	C₂×#3109	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
259	C₂×C₆₇	—	2	4	no	—
260	C₂×C₆₇	—	4	8	no	—
261	C₂×C₆₇	—	8	4	no	—
262	C₂×C₆₇	#3109	1	4	yes	direct product factor
263	C₂×C₆₇	C₂×#525	1	1	yes	characteristic
264	C₂×C₆₇	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
265	C₂×C₆₇	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
266	C₂×#532	C₂×C₆₇	1	1	yes	characteristic, complemented
267	C₂²×C₄×D₄	C₂×C₆₇	1	1	yes	characteristic, complemented
268	#3109	C₂×C₆₇	1	16	yes	direct product factor
269	C₂×#481	C₂×C₆₇	1	2	yes	complemented
270	C₂×#476	C₂×C₆₇	1	2	yes	complemented
271	C₂×#459	C₂×C₆₇	1	2	yes	complemented
272	C₂×#447	C₂×C₆₇	1	2	yes	complemented
273	C₂×#442	C₂×C₆₇	1	4	yes	complemented
274	C₂×#432	C₂×C₆₇	1	1	yes	characteristic, complemented
275	C₆₇	C₂×#3109	1	1	yes	characteristic, direct product factor, Sylow
276	C₂²×#41	—	2	1	no	—
277	C₂³×D₄	C₂²×C₆₇	1	1	yes	characteristic
278	C₂⁴×C₄	—	2	1	no	—
279	C₂²×D₈	C₂²×C₆₇	1	4	yes	—
280	#532	—	2	4	no	—
281	#532	C₂²×C₆₇	1	8	yes	complemented
282	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic, complemented
283	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic, complemented
284	C₂×#124	—	2	1	no	—
285	C₂×#121	—	2	1	no	—
286	C₂×#121	C₂²×C₆₇	1	2	yes	—
287	C₂×C₄×D₄	—	2	1	no	—
288	C₂×C₄×D₄	—	2	1	no	—
289	C₂×C₄×D₄	—	2	4	no	—
290	C₂×C₄×D₄	—	2	1	no	—
291	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
292	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
293	C₂×C₄×D₄	C₂²×C₆₇	1	8	yes	complemented
294	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	complemented
295	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
296	C₂²×C₂²⋊C₄	—	2	1	no	—
297	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
298	#481	C₂²×C₆₇	1	16	yes	complemented
299	#476	C₂²×C₆₇	1	16	yes	complemented
300	#459	C₂²×C₆₇	1	16	yes	complemented
301	#447	C₂²×C₆₇	1	16	yes	complemented
302	#442	C₂²×C₆₇	1	32	yes	complemented
303	#432	C₂²×C₆₇	1	8	yes	complemented
304	C₂×#106	C₂²×C₆₇	1	4	yes	complemented
305	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	complemented
306	C₂×#102	C₂²×C₆₇	1	4	yes	—
307	C₂×#102	C₂²×C₆₇	1	4	yes	complemented
308	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
309	C₂×#97	C₂²×C₆₇	1	2	yes	complemented
310	C₂⁵	—	2	1	no	—
311	C₂×#41	—	2	8	no	—
312	C₂×#41	—	4	8	no	—
313	C₂²×Q₈	—	2	1	no	—
314	C₂²×D₄	—	2	8	no	—
315	C₂²×D₄	—	2	2	no	—
316	C₂²×D₄	—	2	1	no	—
317	C₂²×D₄	—	2	2	no	—
318	C₂²×D₄	—	2	1	no	—
319	C₂²×D₄	—	2	1	no	—
320	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
321	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
322	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
323	C₂³×C₄	—	2	1	no	—
324	C₂³×C₄	—	2	2	no	—
325	C₂³×C₄	—	2	2	no	—
326	C₂³×C₄	—	2	8	no	—
327	C₂³×C₄	—	2	2	no	—
328	C₂³×C₄	—	4	8	no	—
329	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
330	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
331	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
332	C₂×D₈	—	2	16	no	—
333	C₂×D₈	C₂³×C₆₇	1	16	yes	—
334	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
335	C₄⋊Q₈	—	2	2	no	—
336	C₄⋊Q₈	C₂³×C₆₇	1	4	yes	complemented
337	#127	—	2	2	no	—
338	#127	C₂³×C₆₇	1	4	yes	complemented
339	#124	—	2	8	no	—
340	#121	—	2	8	no	—
341	#121	—	2	4	no	—
342	#121	C₂³×C₆₇	1	8	yes	—
343	C₄×D₄	—	2	4	no	—
344	C₄×D₄	—	2	4	no	—
345	C₄×D₄	—	2	8	no	—
346	C₄×D₄	—	2	8	no	—
347	C₄×D₄	—	2	8	no	—
348	C₄×D₄	—	2	4	no	—
349	C₄×D₄	C₂³×C₆₇	1	8	yes	—
350	C₄×D₄	C₂³×C₆₇	1	8	yes	—
351	C₄×D₄	C₂³×C₆₇	1	8	yes	complemented
352	C₂×C₄⋊C₄	—	2	1	no	—
353	C₂×C₄⋊C₄	—	2	2	no	—
354	C₂×C₄⋊C₄	—	2	1	no	—
355	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
356	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
357	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
358	C₂×C₂²⋊C₄	—	2	2	no	—
359	C₂×C₂²⋊C₄	—	2	2	no	—
360	C₂×C₂²⋊C₄	—	2	8	no	—
361	C₂×C₂²⋊C₄	—	2	2	no	—
362	C₂×C₂²⋊C₄	—	2	2	no	—
363	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
364	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
365	C₂×C₄²	D₄×C₆₇	1	4	yes	—
366	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
367	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
368	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
369	#106	C₂³×C₆₇	1	16	yes	complemented
370	C₄⋊C₈	C₂³×C₆₇	1	8	yes	complemented
371	#102	C₂³×C₆₇	1	16	yes	—
372	#102	C₂³×C₆₇	1	16	yes	complemented
373	C₂²⋊C₈	C₂³×C₆₇	1	8	yes	—
374	#97	C₂³×C₆₇	1	8	yes	complemented
375	C₂⁴	—	2	8	no	—
376	C₂⁴	—	2	4	no	—
377	C₂⁴	—	2	2	no	—
378	C₂⁴	—	2	2	no	—
379	C₂⁴	—	4	4	no	—
380	C₂⁴	—	4	4	no	—
381	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
382	#41	—	4	32	no	—
383	C₂×Q₈	—	2	4	no	—
384	C₂×Q₈	—	4	4	no	—
385	C₂×D₄	—	2	8	no	—
386	C₂×D₄	—	2	16	no	—
387	C₂×D₄	—	2	16	no	—
388	C₂×D₄	—	2	8	no	—
389	C₂×D₄	—	2	4	no	—
390	C₂×D₄	—	2	2	no	—
391	C₂×D₄	—	2	2	no	—
392	C₂×D₄	—	2	4	no	—
393	C₂×D₄	—	2	4	no	—
394	C₂×D₄	—	2	4	no	—
395	C₂×D₄	—	2	8	no	—
396	C₂×D₄	—	4	4	no	—
397	C₂×D₄	—	4	8	no	—
398	C₂×D₄	—	4	4	no	—
399	C₂×D₄	#41×C₆₇	1	8	yes	complemented
400	C₂×D₄	C₂⁴×C₆₇	1	4	yes	—
401	C₂×D₄	C₂⁴×C₆₇	1	4	yes	—
402	C₂²×C₄	—	2	4	no	—
403	C₂²×C₄	—	2	4	no	—
404	C₂²×C₄	—	2	4	no	—
405	C₂²×C₄	—	2	4	no	—
406	C₂²×C₄	—	2	2	no	—
407	C₂²×C₄	—	2	2	no	—
408	C₂²×C₄	—	2	8	no	—
409	C₂²×C₄	—	2	8	no	—
410	C₂²×C₄	—	2	8	no	—
411	C₂²×C₄	—	4	8	no	—
412	C₂²×C₄	—	4	32	no	—
413	C₂²×C₄	—	4	8	no	—
414	C₂²×C₄	—	4	4	no	—
415	C₂²×C₄	—	4	8	no	—
416	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
417	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
418	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
419	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
420	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
421	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
422	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
423	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
424	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
425	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
426	D₈	—	2	32	no	—
427	C₂×C₈	—	2	8	no	—
428	C₂×C₈	C₂⁴×C₆₇	1	8	yes	—
429	C₄⋊C₄	—	2	4	no	—
430	C₄⋊C₄	—	2	2	no	—
431	C₄⋊C₄	—	2	4	no	—
432	C₄⋊C₄	—	2	2	no	—
433	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
434	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
435	C₂²⋊C₄	—	2	8	no	—
436	C₂²⋊C₄	—	2	8	no	—
437	C₂²⋊C₄	—	2	8	no	—
438	C₂²⋊C₄	—	2	8	no	—
439	C₄²	C₂×D₄×C₆₇	1	4	yes	—
440	C₄²	C₂×D₄×C₆₇	1	2	yes	—
441	C₄²	C₂×D₄×C₆₇	1	4	yes	—
442	C₄²	C₂×D₄×C₆₇	1	2	yes	—
443	C₄²	C₂⁴×C₆₇	1	2	yes	—
444	C₄²	C₂⁴×C₆₇	1	2	yes	—
445	C₂³	—	2	2	no	—
446	C₂³	—	2	2	no	—
447	C₂³	—	2	16	no	—
448	C₂³	—	2	8	no	—
449	C₂³	—	2	4	no	—
450	C₂³	—	2	4	no	—
451	C₂³	—	4	8	no	—
452	C₂³	—	4	32	no	—
453	C₂³	—	4	4	no	—
454	C₂³	—	4	8	no	—
455	C₂³	—	4	4	no	—
456	C₂³	—	4	4	no	—
457	C₂³	—	4	4	no	—
458	C₂³	—	4	4	no	—
459	C₂³	AGL(1,ℤ/8)×C₆₇	1	4	yes	—
460	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
461	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
462	Q₈	—	4	8	no	—
463	D₄	—	2	16	no	—
464	D₄	—	2	8	no	—
465	D₄	—	2	4	no	—
466	D₄	—	2	8	no	—
467	D₄	—	4	8	no	—
468	D₄	—	4	16	no	—
469	D₄	—	4	4	no	—
470	D₄	—	4	8	no	—
471	D₄	C₂×#41×C₆₇	1	8	yes	complemented
472	C₂×C₄	—	2	2	no	—
473	C₂×C₄	—	2	2	no	—
474	C₂×C₄	—	2	2	no	—
475	C₂×C₄	—	2	2	no	—
476	C₂×C₄	—	2	8	no	—
477	C₂×C₄	—	2	4	no	—
478	C₂×C₄	—	2	8	no	—
479	C₂×C₄	—	2	4	no	—
480	C₂×C₄	—	2	4	no	—
481	C₂×C₄	—	2	4	no	—
482	C₂×C₄	—	4	16	no	—
483	C₂×C₄	—	4	16	no	—
484	C₂×C₄	—	4	8	no	—
485	C₂×C₄	—	4	4	no	—
486	C₂×C₄	—	4	4	no	—
487	C₂×C₄	—	4	16	no	—
488	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	4	yes	—
489	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
490	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
491	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
492	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
493	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
494	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
495	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
496	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
497	C₈	—	2	8	no	—
498	C₂²	—	2	2	no	—
499	C₂²	—	2	8	no	—
500	C₂²	—	2	2	no	—
501	C₂²	—	2	8	no	—
502	C₂²	—	4	32	no	—
503	C₂²	—	4	4	no	—
504	C₂²	—	4	4	no	—
505	C₂²	—	4	16	no	—
506	C₂²	—	4	4	no	—
507	C₂²	—	4	4	no	—
508	C₂²	—	4	4	no	—
509	C₂²	—	8	8	no	—
510	C₂²	#525×C₆₇	1	2	yes	—
511	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	4	yes	—
512	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	2	yes	—
513	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	2	yes	—
514	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
515	C₄	—	2	2	no	—
516	C₄	—	2	2	no	—
517	C₄	—	2	2	no	—
518	C₄	—	2	2	no	—
519	C₄	—	4	4	no	—
520	C₄	—	4	4	no	—
521	C₄	#525×C₆₇	1	2	yes	—
522	C₄	C₂×AGL(1,ℤ/8)×C₆₇	1	4	yes	—
523	C₄	C₂²×#41×C₆₇	1	2	yes	—
524	C₂	—	2	4	no	—
525	C₂	—	4	8	no	—
526	C₂	—	8	4	no	—
527	C₂	C₆₇×#3109	1	4	yes	direct product factor
528	C₂	C₂×#525×C₆₇	1	1	yes	characteristic
529	C₂	C₂²×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
530	C₂	C₂²×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂²×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#532×C₆₇	C₂	semidirect
2. C₂²×C₄×D₄×C₆₇	C₂	semidirect
3. C₆₇×#3109	C₂	direct (N × H)
4. C₂×#481×C₆₇	C₂	semidirect
5. C₂×#476×C₆₇	C₂	semidirect
6. C₂×#459×C₆₇	C₂	semidirect
7. C₂×#447×C₆₇	C₂	semidirect
8. C₂×#442×C₆₇	C₂	semidirect
9. C₂×#432×C₆₇	C₂	semidirect
15. #532×C₆₇	C₂²	semidirect
16. C₂×C₄⋊Q₈×C₆₇	C₂²	semidirect
17. C₂×#127×C₆₇	C₂²	semidirect
27. C₂×C₄×D₄×C₆₇	C₂²	semidirect
28. C₂×C₄×D₄×C₆₇	C₂²	semidirect
32. #481×C₆₇	C₂²	semidirect
33. #476×C₆₇	C₂²	semidirect
34. #459×C₆₇	C₂²	semidirect
35. #447×C₆₇	C₂²	semidirect
36. #442×C₆₇	C₂²	semidirect
37. #432×C₆₇	C₂²	semidirect
38. C₂×#106×C₆₇	C₂²	semidirect
39. C₂×C₄⋊C₈×C₆₇	C₂²	semidirect
41. C₂×#102×C₆₇	C₂²	semidirect
43. C₂×#97×C₆₇	C₂²	semidirect
70. C₄⋊Q₈×C₆₇	C₂³	semidirect
72. #127×C₆₇	C₂³	semidirect
85. C₄×D₄×C₆₇	C₂³	semidirect
103. #106×C₆₇	C₂³	semidirect
104. C₄⋊C₈×C₆₇	C₂³	semidirect
106. #102×C₆₇	C₂³	semidirect
108. #97×C₆₇	C₂³	semidirect
133. C₂×D₄×C₆₇	#41	semidirect
205. D₄×C₆₇	C₂×#41	semidirect
258. C₂×#3109	C₆₇	direct (N × H)
262. C₂×C₆₇	#3109	direct (N × H)
266. C₂×#532	C₂×C₆₇	semidirect
267. C₂²×C₄×D₄	C₂×C₆₇	semidirect
268. #3109	C₂×C₆₇	direct (N × H)
269. C₂×#481	C₂×C₆₇	semidirect
270. C₂×#476	C₂×C₆₇	semidirect
271. C₂×#459	C₂×C₆₇	semidirect
272. C₂×#447	C₂×C₆₇	semidirect
273. C₂×#442	C₂×C₆₇	semidirect
274. C₂×#432	C₂×C₆₇	semidirect
275. C₆₇	C₂×#3109	direct (N × H)
281. #532	C₂²×C₆₇	semidirect
282. C₂×C₄⋊Q₈	C₂²×C₆₇	semidirect
283. C₂×#127	C₂²×C₆₇	semidirect
293. C₂×C₄×D₄	C₂²×C₆₇	semidirect
294. C₂×C₄×D₄	C₂²×C₆₇	semidirect
298. #481	C₂²×C₆₇	semidirect
299. #476	C₂²×C₆₇	semidirect
300. #459	C₂²×C₆₇	semidirect
301. #447	C₂²×C₆₇	semidirect
302. #442	C₂²×C₆₇	semidirect
303. #432	C₂²×C₆₇	semidirect
304. C₂×#106	C₂²×C₆₇	semidirect
305. C₂×C₄⋊C₈	C₂²×C₆₇	semidirect
307. C₂×#102	C₂²×C₆₇	semidirect
309. C₂×#97	C₂²×C₆₇	semidirect
336. C₄⋊Q₈	C₂³×C₆₇	semidirect
338. #127	C₂³×C₆₇	semidirect
351. C₄×D₄	C₂³×C₆₇	semidirect
369. #106	C₂³×C₆₇	semidirect
370. C₄⋊C₈	C₂³×C₆₇	semidirect
372. #102	C₂³×C₆₇	semidirect
374. #97	C₂³×C₆₇	semidirect
399. C₂×D₄	#41×C₆₇	semidirect
471. D₄	C₂×#41×C₆₇	semidirect
527. C₂	C₆₇×#3109	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#532×C₆₇	2
C₂²×C₄×D₄×C₆₇	2
C₆₇×#3109	2
C₂×#481×C₆₇	2
C₂×#476×C₆₇	2
C₂×#459×C₆₇	2
C₂×#447×C₆₇	2
C₂×#442×C₆₇	2
C₂×#432×C₆₇	2
C₂×#3109	67

Subgroups — C2×C67×#3109 #30739792

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#3109 #30739792