Subgroups — C₂×C₆₇×#3339 #30740481

Order 17152.

1994 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#553×C₆₇	C₂	1	2	yes	maximal, complemented
2	C₆₇×#3339	C₂	1	16	yes	maximal, direct product factor
3	C₂×#499×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#481×C₆₇	C₂	1	2	yes	maximal, complemented
5	C₂×#482×C₆₇	C₂	1	2	yes	maximal, complemented
6	C₂×#478×C₆₇	C₂	1	2	yes	maximal, complemented
7	C₂×#466×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#466×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₄×D₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₈×D₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#427×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#107×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×D₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
16	#553×C₆₇	—	2	8	no	—
17	#553×C₆₇	C₂²	1	16	yes	complemented
18	C₂×#126×C₆₇	—	2	2	no	—
19	C₂×#125×C₆₇	C₂²	1	2	yes	—
20	C₂×#123×C₆₇	—	2	2	no	—
21	C₂×#121×C₆₇	—	2	2	no	—
22	C₂×#121×C₆₇	C₂²	1	2	yes	—
23	C₂×#121×C₆₇	C₂²	1	2	yes	—
24	C₂×C₄×D₄×C₆₇	—	2	2	no	—
25	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
26	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
27	C₂×#117×C₆₇	C₂²	1	2	yes	—
28	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
29	#499×C₆₇	C₂²	1	8	yes	complemented
30	#482×C₆₇	C₂²	1	16	yes	complemented
31	#481×C₆₇	C₂²	1	16	yes	complemented
32	#478×C₆₇	C₂²	1	16	yes	complemented
33	#466×C₆₇	C₂²	1	8	yes	complemented
34	#466×C₆₇	C₂²	1	8	yes	complemented
35	C₄×D₈×C₆₇	C₂²	1	8	yes	complemented
36	C₈×D₄×C₆₇	C₂²	1	8	yes	complemented
37	#427×C₆₇	C₂²	1	8	yes	complemented
38	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
39	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
40	C₂×#107×C₆₇	C₂²	1	2	yes	—
41	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
42	C₂×#107×C₆₇	C₂²	1	8	yes	complemented
43	C₂×#106×C₆₇	C₂²	1	2	yes	—
44	C₂×#106×C₆₇	C₂²	1	2	yes	—
45	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
46	C₂×#102×C₆₇	C₂²	1	2	yes	—
47	C₂×#102×C₆₇	C₂²	1	2	yes	—
48	C₂×#102×C₆₇	C₂²	1	2	yes	—
49	C₂×C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
50	C₂×C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
51	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
52	C₂²×D₄×C₆₇	—	2	2	no	—
53	C₂²×D₄×C₆₇	—	2	2	no	—
54	C₂²×D₄×C₆₇	D₄	1	1	yes	characteristic
55	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
56	C₂³×C₄×C₆₇	—	2	2	no	—
57	C₂³×C₄×C₆₇	—	2	2	no	—
58	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
59	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
60	C₂×D₈×C₆₇	—	2	4	no	—
61	C₂×D₈×C₆₇	C₂³	1	4	yes	—
62	C₂²×C₈×C₆₇	—	2	4	no	—
63	C₂²×C₈×C₆₇	—	2	4	no	—
64	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
65	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
66	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
67	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
68	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
69	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
70	#126×C₆₇	—	2	16	no	—
71	#125×C₆₇	—	2	4	no	—
72	#125×C₆₇	C₂³	1	8	yes	—
73	#123×C₆₇	—	2	16	no	—
74	#121×C₆₇	—	2	16	no	—
75	#121×C₆₇	—	2	4	no	—
76	#121×C₆₇	—	2	4	no	—
77	#121×C₆₇	C₂³	1	8	yes	—
78	#121×C₆₇	C₂³	1	8	yes	—
79	C₄×D₄×C₆₇	—	2	16	no	—
80	C₄×D₄×C₆₇	—	2	4	no	—
81	C₄×D₄×C₆₇	D₄	1	4	yes	—
82	C₄×D₄×C₆₇	C₂³	1	8	yes	—
83	C₄×D₄×C₆₇	C₂³	1	4	yes	—
84	#117×C₆₇	—	2	4	no	—
85	#117×C₆₇	C₂³	1	8	yes	—
86	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
87	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
88	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
89	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
90	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
91	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
92	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
93	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
94	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
95	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
96	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
97	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
98	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
99	C₂×C₂²⋊C₄×C₆₇	D₄	1	1	yes	characteristic
100	C₂×C₂²⋊C₄×C₆₇	D₄	1	1	yes	characteristic
101	C₂×C₄²×C₆₇	—	2	2	no	—
102	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
103	#107×C₆₇	C₂³	1	4	yes	—
104	#107×C₆₇	C₂³	1	4	yes	—
105	#107×C₆₇	C₂³	1	8	yes	—
106	#107×C₆₇	C₂³	1	4	yes	—
107	#106×C₆₇	C₂³	1	8	yes	—
108	#106×C₆₇	C₂³	1	8	yes	—
109	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
110	#102×C₆₇	C₂³	1	8	yes	—
111	#102×C₆₇	C₂³	1	8	yes	—
112	#102×C₆₇	C₂³	1	8	yes	—
113	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
114	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
115	C₄×C₈×C₆₇	C₂³	1	4	yes	—
116	C₂⁴×C₆₇	—	2	2	no	—
117	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
118	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
119	C₂×D₄×C₆₇	—	2	4	no	—
120	C₂×D₄×C₆₇	—	2	8	no	—
121	C₂×D₄×C₆₇	—	2	8	no	—
122	C₂×D₄×C₆₇	—	2	2	no	—
123	C₂×D₄×C₆₇	—	2	4	no	—
124	C₂×D₄×C₆₇	—	2	4	no	—
125	C₂×D₄×C₆₇	—	4	8	no	—
126	C₂×D₄×C₆₇	—	8	4	no	—
127	C₂×D₄×C₆₇	C₂×D₄	1	4	yes	—
128	C₂×D₄×C₆₇	C₂⁴	1	4	yes	—
129	C₂²×C₄×C₆₇	—	2	2	no	—
130	C₂²×C₄×C₆₇	—	2	2	no	—
131	C₂²×C₄×C₆₇	—	2	2	no	—
132	C₂²×C₄×C₆₇	—	2	2	no	—
133	C₂²×C₄×C₆₇	—	2	2	no	—
134	C₂²×C₄×C₆₇	—	2	2	no	—
135	C₂²×C₄×C₆₇	—	2	2	no	—
136	C₂²×C₄×C₆₇	—	2	8	no	—
137	C₂²×C₄×C₆₇	—	2	8	no	—
138	C₂²×C₄×C₆₇	—	8	4	no	—
139	C₂²×C₄×C₆₇	—	8	4	no	—
140	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
141	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
142	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
143	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
144	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
145	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
146	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
147	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
148	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
149	D₈×C₆₇	—	2	8	no	—
150	C₂×C₈×C₆₇	—	2	2	no	—
151	C₂×C₈×C₆₇	—	2	2	no	—
152	C₂×C₈×C₆₇	—	2	2	no	—
153	C₂×C₈×C₆₇	—	2	8	no	—
154	C₂×C₈×C₆₇	—	2	8	no	—
155	C₂×C₈×C₆₇	#41	1	4	yes	—
156	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
157	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
158	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
159	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
160	C₄⋊C₄×C₆₇	—	2	2	no	—
161	C₄⋊C₄×C₆₇	—	2	2	no	—
162	C₄⋊C₄×C₆₇	—	2	8	no	—
163	C₄⋊C₄×C₆₇	—	2	8	no	—
164	C₄⋊C₄×C₆₇	—	2	8	no	—
165	C₄⋊C₄×C₆₇	—	2	2	no	—
166	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
167	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
168	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
169	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
170	C₂²⋊C₄×C₆₇	—	2	8	no	—
171	C₂²⋊C₄×C₆₇	—	2	8	no	—
172	C₂²⋊C₄×C₆₇	—	2	8	no	—
173	C₂²⋊C₄×C₆₇	—	2	8	no	—
174	C₂²⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
175	C₂²⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
176	C₄²×C₆₇	—	2	8	no	—
177	C₄²×C₆₇	C₂×D₄	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	2	no	—
182	C₂³×C₆₇	—	2	2	no	—
183	C₂³×C₆₇	—	2	2	no	—
184	C₂³×C₆₇	—	2	4	no	—
185	C₂³×C₆₇	—	4	4	no	—
186	C₂³×C₆₇	—	4	4	no	—
187	C₂³×C₆₇	—	4	4	no	—
188	C₂³×C₆₇	#135	1	4	yes	—
189	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
190	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
191	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
192	D₄×C₆₇	—	2	8	no	—
193	D₄×C₆₇	—	2	4	no	—
194	D₄×C₆₇	—	2	4	no	—
195	D₄×C₆₇	—	4	16	no	—
196	D₄×C₆₇	—	4	4	no	—
197	D₄×C₆₇	—	8	8	no	—
198	C₂×C₄×C₆₇	—	2	2	no	—
199	C₂×C₄×C₆₇	—	2	2	no	—
200	C₂×C₄×C₆₇	—	2	2	no	—
201	C₂×C₄×C₆₇	—	2	2	no	—
202	C₂×C₄×C₆₇	—	2	4	no	—
203	C₂×C₄×C₆₇	—	2	4	no	—
204	C₂×C₄×C₆₇	—	2	2	no	—
205	C₂×C₄×C₆₇	—	2	2	no	—
206	C₂×C₄×C₆₇	—	2	2	no	—
207	C₂×C₄×C₆₇	—	2	2	no	—
208	C₂×C₄×C₆₇	—	2	4	no	—
209	C₂×C₄×C₆₇	—	2	4	no	—
210	C₂×C₄×C₆₇	—	2	4	no	—
211	C₂×C₄×C₆₇	—	4	4	no	—
212	C₂×C₄×C₆₇	—	4	4	no	—
213	C₂×C₄×C₆₇	—	4	4	no	—
214	C₂×C₄×C₆₇	—	8	8	no	—
215	C₂×C₄×C₆₇	—	8	8	no	—
216	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
217	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
218	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
219	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
220	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
221	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
222	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
223	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
224	C₈×C₆₇	—	2	2	no	—
225	C₈×C₆₇	—	2	2	no	—
226	C₈×C₆₇	—	2	2	no	—
227	C₈×C₆₇	C₂×#41	1	4	yes	—
228	C₂²×C₆₇	—	2	8	no	—
229	C₂²×C₆₇	—	2	2	no	—
230	C₂²×C₆₇	—	2	2	no	—
231	C₂²×C₆₇	—	2	2	no	—
232	C₂²×C₆₇	—	2	2	no	—
233	C₂²×C₆₇	—	2	2	no	—
234	C₂²×C₆₇	—	4	4	no	—
235	C₂²×C₆₇	—	4	4	no	—
236	C₂²×C₆₇	—	4	4	no	—
237	C₂²×C₆₇	—	4	4	no	—
238	C₂²×C₆₇	—	8	8	no	—
239	C₂²×C₆₇	#547	1	2	yes	—
240	C₂²×C₆₇	C₂×#135	1	4	yes	—
241	C₂²×C₆₇	C₂×#135	1	2	yes	—
242	C₂²×C₆₇	#576	1	2	yes	—
243	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
244	C₄×C₆₇	—	2	2	no	—
245	C₄×C₆₇	—	2	2	no	—
246	C₄×C₆₇	—	2	2	no	—
247	C₄×C₆₇	—	2	2	no	—
248	C₄×C₆₇	—	4	4	no	—
249	C₄×C₆₇	—	4	4	no	—
250	C₄×C₆₇	—	4	4	no	—
251	C₄×C₆₇	#547	1	2	yes	—
252	C₄×C₆₇	C₂²×#41	1	1	yes	characteristic
253	C₄×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
254	C₂×#3339	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
255	C₂×C₆₇	—	2	4	no	—
256	C₂×C₆₇	—	4	2	no	—
257	C₂×C₆₇	—	8	4	no	—
258	C₂×C₆₇	#3339	1	4	yes	direct product factor
259	C₂×C₆₇	C₂×#547	1	1	yes	characteristic
260	C₂×C₆₇	C₂²×#135	1	1	yes	characteristic
261	C₂×C₆₇	C₂×#576	1	1	yes	characteristic
262	C₂×#553	C₂×C₆₇	1	2	yes	complemented
263	#3339	C₂×C₆₇	1	16	yes	direct product factor
264	C₂×#499	C₂×C₆₇	1	1	yes	characteristic, complemented
265	C₂×#481	C₂×C₆₇	1	2	yes	complemented
266	C₂×#482	C₂×C₆₇	1	2	yes	complemented
267	C₂×#478	C₂×C₆₇	1	2	yes	complemented
268	C₂×#466	C₂×C₆₇	1	1	yes	characteristic, complemented
269	C₂×#466	C₂×C₆₇	1	1	yes	characteristic, complemented
270	C₂×C₄×D₈	C₂×C₆₇	1	1	yes	characteristic, complemented
271	C₂×C₈×D₄	C₂×C₆₇	1	1	yes	characteristic, complemented
272	C₂×#427	C₂×C₆₇	1	1	yes	characteristic, complemented
273	C₂²×#107	C₂×C₆₇	1	1	yes	characteristic, complemented
274	C₆₇	C₂×#3339	1	1	yes	characteristic, direct product factor, Sylow
275	C₂²×D₈	C₂²×C₆₇	1	1	yes	characteristic
276	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
277	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
278	#553	—	2	8	no	—
279	#553	C₂²×C₆₇	1	16	yes	complemented
280	C₂×#126	—	2	2	no	—
281	C₂×#125	C₂²×C₆₇	1	2	yes	—
282	C₂×#123	—	2	2	no	—
283	C₂×#121	—	2	2	no	—
284	C₂×#121	C₂²×C₆₇	1	2	yes	—
285	C₂×#121	C₂²×C₆₇	1	2	yes	—
286	C₂×C₄×D₄	—	2	2	no	—
287	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
288	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
289	C₂×#117	C₂²×C₆₇	1	2	yes	—
290	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
291	#499	C₂²×C₆₇	1	8	yes	complemented
292	#482	C₂²×C₆₇	1	16	yes	complemented
293	#481	C₂²×C₆₇	1	16	yes	complemented
294	#478	C₂²×C₆₇	1	16	yes	complemented
295	#466	C₂²×C₆₇	1	8	yes	complemented
296	#466	C₂²×C₆₇	1	8	yes	complemented
297	C₄×D₈	C₂²×C₆₇	1	8	yes	complemented
298	C₈×D₄	C₂²×C₆₇	1	8	yes	complemented
299	#427	C₂²×C₆₇	1	8	yes	complemented
300	C₂×#107	C₂²×C₆₇	1	2	yes	—
301	C₂×#107	C₂²×C₆₇	1	1	yes	characteristic
302	C₂×#107	C₂²×C₆₇	1	1	yes	characteristic
303	C₂×#107	C₂²×C₆₇	1	1	yes	characteristic
304	C₂×#107	C₂²×C₆₇	1	8	yes	complemented
305	C₂×#106	C₂²×C₆₇	1	2	yes	—
306	C₂×#106	C₂²×C₆₇	1	2	yes	—
307	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
308	C₂×#102	C₂²×C₆₇	1	2	yes	—
309	C₂×#102	C₂²×C₆₇	1	2	yes	—
310	C₂×#102	C₂²×C₆₇	1	2	yes	—
311	C₂×C₂²⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
312	C₂×C₂²⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
313	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
314	C₂²×D₄	—	2	2	no	—
315	C₂²×D₄	—	2	2	no	—
316	C₂²×D₄	D₄×C₆₇	1	1	yes	characteristic
317	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
318	C₂³×C₄	—	2	2	no	—
319	C₂³×C₄	—	2	2	no	—
320	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
321	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
322	C₂×D₈	—	2	4	no	—
323	C₂×D₈	C₂³×C₆₇	1	4	yes	—
324	C₂²×C₈	—	2	4	no	—
325	C₂²×C₈	—	2	4	no	—
326	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
327	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
328	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
329	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
330	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
331	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
332	#126	—	2	16	no	—
333	#125	—	2	4	no	—
334	#125	C₂³×C₆₇	1	8	yes	—
335	#123	—	2	16	no	—
336	#121	—	2	16	no	—
337	#121	—	2	4	no	—
338	#121	—	2	4	no	—
339	#121	C₂³×C₆₇	1	8	yes	—
340	#121	C₂³×C₆₇	1	8	yes	—
341	C₄×D₄	—	2	16	no	—
342	C₄×D₄	—	2	4	no	—
343	C₄×D₄	D₄×C₆₇	1	4	yes	—
344	C₄×D₄	C₂³×C₆₇	1	8	yes	—
345	C₄×D₄	C₂³×C₆₇	1	4	yes	—
346	#117	—	2	4	no	—
347	#117	C₂³×C₆₇	1	8	yes	—
348	C₂×C₄⋊C₄	—	2	2	no	—
349	C₂×C₄⋊C₄	—	2	4	no	—
350	C₂×C₄⋊C₄	—	2	2	no	—
351	C₂×C₄⋊C₄	—	2	2	no	—
352	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
353	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
354	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
355	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
356	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
357	C₂×C₂²⋊C₄	—	2	2	no	—
358	C₂×C₂²⋊C₄	—	2	2	no	—
359	C₂×C₂²⋊C₄	—	2	2	no	—
360	C₂×C₂²⋊C₄	—	2	2	no	—
361	C₂×C₂²⋊C₄	D₄×C₆₇	1	1	yes	characteristic
362	C₂×C₂²⋊C₄	D₄×C₆₇	1	1	yes	characteristic
363	C₂×C₄²	—	2	2	no	—
364	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
365	#107	C₂³×C₆₇	1	4	yes	—
366	#107	C₂³×C₆₇	1	4	yes	—
367	#107	C₂³×C₆₇	1	8	yes	—
368	#107	C₂³×C₆₇	1	4	yes	—
369	#106	C₂³×C₆₇	1	8	yes	—
370	#106	C₂³×C₆₇	1	8	yes	—
371	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
372	#102	C₂³×C₆₇	1	8	yes	—
373	#102	C₂³×C₆₇	1	8	yes	—
374	#102	C₂³×C₆₇	1	8	yes	—
375	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
376	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
377	C₄×C₈	C₂³×C₆₇	1	4	yes	—
378	C₂⁴	—	2	2	no	—
379	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
380	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
381	C₂×D₄	—	2	4	no	—
382	C₂×D₄	—	2	8	no	—
383	C₂×D₄	—	2	8	no	—
384	C₂×D₄	—	2	2	no	—
385	C₂×D₄	—	2	4	no	—
386	C₂×D₄	—	2	4	no	—
387	C₂×D₄	—	4	8	no	—
388	C₂×D₄	—	8	4	no	—
389	C₂×D₄	C₂×D₄×C₆₇	1	4	yes	—
390	C₂×D₄	C₂⁴×C₆₇	1	4	yes	—
391	C₂²×C₄	—	2	2	no	—
392	C₂²×C₄	—	2	2	no	—
393	C₂²×C₄	—	2	2	no	—
394	C₂²×C₄	—	2	2	no	—
395	C₂²×C₄	—	2	2	no	—
396	C₂²×C₄	—	2	2	no	—
397	C₂²×C₄	—	2	2	no	—
398	C₂²×C₄	—	2	8	no	—
399	C₂²×C₄	—	2	8	no	—
400	C₂²×C₄	—	8	4	no	—
401	C₂²×C₄	—	8	4	no	—
402	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
403	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
404	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
405	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
406	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
407	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
408	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
409	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
410	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
411	D₈	—	2	8	no	—
412	C₂×C₈	—	2	2	no	—
413	C₂×C₈	—	2	2	no	—
414	C₂×C₈	—	2	2	no	—
415	C₂×C₈	—	2	8	no	—
416	C₂×C₈	—	2	8	no	—
417	C₂×C₈	#41×C₆₇	1	4	yes	—
418	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
419	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
420	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
421	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
422	C₄⋊C₄	—	2	2	no	—
423	C₄⋊C₄	—	2	2	no	—
424	C₄⋊C₄	—	2	8	no	—
425	C₄⋊C₄	—	2	8	no	—
426	C₄⋊C₄	—	2	8	no	—
427	C₄⋊C₄	—	2	2	no	—
428	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
429	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
430	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
431	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
432	C₂²⋊C₄	—	2	8	no	—
433	C₂²⋊C₄	—	2	8	no	—
434	C₂²⋊C₄	—	2	8	no	—
435	C₂²⋊C₄	—	2	8	no	—
436	C₂²⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
437	C₂²⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
438	C₄²	—	2	8	no	—
439	C₄²	C₂×D₄×C₆₇	1	2	yes	—
440	C₄²	C₂⁴×C₆₇	1	2	yes	—
441	C₂³	—	2	2	no	—
442	C₂³	—	2	2	no	—
443	C₂³	—	2	2	no	—
444	C₂³	—	2	2	no	—
445	C₂³	—	2	2	no	—
446	C₂³	—	2	4	no	—
447	C₂³	—	4	4	no	—
448	C₂³	—	4	4	no	—
449	C₂³	—	4	4	no	—
450	C₂³	#135×C₆₇	1	4	yes	—
451	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
452	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
453	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
454	D₄	—	2	8	no	—
455	D₄	—	2	4	no	—
456	D₄	—	2	4	no	—
457	D₄	—	4	16	no	—
458	D₄	—	4	4	no	—
459	D₄	—	8	8	no	—
460	C₂×C₄	—	2	2	no	—
461	C₂×C₄	—	2	2	no	—
462	C₂×C₄	—	2	2	no	—
463	C₂×C₄	—	2	2	no	—
464	C₂×C₄	—	2	4	no	—
465	C₂×C₄	—	2	4	no	—
466	C₂×C₄	—	2	2	no	—
467	C₂×C₄	—	2	2	no	—
468	C₂×C₄	—	2	2	no	—
469	C₂×C₄	—	2	2	no	—
470	C₂×C₄	—	2	4	no	—
471	C₂×C₄	—	2	4	no	—
472	C₂×C₄	—	2	4	no	—
473	C₂×C₄	—	4	4	no	—
474	C₂×C₄	—	4	4	no	—
475	C₂×C₄	—	4	4	no	—
476	C₂×C₄	—	8	8	no	—
477	C₂×C₄	—	8	8	no	—
478	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
479	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
480	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
481	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
482	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
483	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
484	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
485	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
486	C₈	—	2	2	no	—
487	C₈	—	2	2	no	—
488	C₈	—	2	2	no	—
489	C₈	C₂×#41×C₆₇	1	4	yes	—
490	C₂²	—	2	2	no	—
491	C₂²	—	2	8	no	—
492	C₂²	—	2	2	no	—
493	C₂²	—	2	2	no	—
494	C₂²	—	2	2	no	—
495	C₂²	—	2	2	no	—
496	C₂²	—	4	4	no	—
497	C₂²	—	4	4	no	—
498	C₂²	—	4	4	no	—
499	C₂²	—	4	4	no	—
500	C₂²	—	8	8	no	—
501	C₂²	#547×C₆₇	1	2	yes	—
502	C₂²	C₂×#135×C₆₇	1	4	yes	—
503	C₂²	C₂×#135×C₆₇	1	2	yes	—
504	C₂²	#576×C₆₇	1	2	yes	—
505	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
506	C₄	—	2	2	no	—
507	C₄	—	2	2	no	—
508	C₄	—	2	2	no	—
509	C₄	—	2	2	no	—
510	C₄	—	4	4	no	—
511	C₄	—	4	4	no	—
512	C₄	—	4	4	no	—
513	C₄	#547×C₆₇	1	2	yes	—
514	C₄	C₂²×#41×C₆₇	1	1	yes	characteristic
515	C₄	C₂×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
516	C₂	—	2	4	no	—
517	C₂	—	4	2	no	—
518	C₂	—	8	4	no	—
519	C₂	C₆₇×#3339	1	4	yes	direct product factor
520	C₂	C₂×#547×C₆₇	1	1	yes	characteristic
521	C₂	C₂²×#135×C₆₇	1	1	yes	characteristic
522	C₂	C₂×#576×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 ◁ C₂×C₄×D₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#553×C₆₇	C₂	semidirect
2. C₆₇×#3339	C₂	direct (N × H)
3. C₂×#499×C₆₇	C₂	semidirect
4. C₂×#481×C₆₇	C₂	semidirect
5. C₂×#482×C₆₇	C₂	semidirect
6. C₂×#478×C₆₇	C₂	semidirect
7. C₂×#466×C₆₇	C₂	semidirect
8. C₂×#466×C₆₇	C₂	semidirect
9. C₂×C₄×D₈×C₆₇	C₂	semidirect
10. C₂×C₈×D₄×C₆₇	C₂	semidirect
11. C₂×#427×C₆₇	C₂	semidirect
12. C₂²×#107×C₆₇	C₂	semidirect
17. #553×C₆₇	C₂²	semidirect
29. #499×C₆₇	C₂²	semidirect
30. #482×C₆₇	C₂²	semidirect
31. #481×C₆₇	C₂²	semidirect
32. #478×C₆₇	C₂²	semidirect
33. #466×C₆₇	C₂²	semidirect
34. #466×C₆₇	C₂²	semidirect
35. C₄×D₈×C₆₇	C₂²	semidirect
36. C₈×D₄×C₆₇	C₂²	semidirect
37. #427×C₆₇	C₂²	semidirect
42. C₂×#107×C₆₇	C₂²	semidirect
254. C₂×#3339	C₆₇	direct (N × H)
258. C₂×C₆₇	#3339	direct (N × H)
262. C₂×#553	C₂×C₆₇	semidirect
263. #3339	C₂×C₆₇	direct (N × H)
264. C₂×#499	C₂×C₆₇	semidirect
265. C₂×#481	C₂×C₆₇	semidirect
266. C₂×#482	C₂×C₆₇	semidirect
267. C₂×#478	C₂×C₆₇	semidirect
268. C₂×#466	C₂×C₆₇	semidirect
269. C₂×#466	C₂×C₆₇	semidirect
270. C₂×C₄×D₈	C₂×C₆₇	semidirect
271. C₂×C₈×D₄	C₂×C₆₇	semidirect
272. C₂×#427	C₂×C₆₇	semidirect
273. C₂²×#107	C₂×C₆₇	semidirect
274. C₆₇	C₂×#3339	direct (N × H)
279. #553	C₂²×C₆₇	semidirect
291. #499	C₂²×C₆₇	semidirect
292. #482	C₂²×C₆₇	semidirect
293. #481	C₂²×C₆₇	semidirect
294. #478	C₂²×C₆₇	semidirect
295. #466	C₂²×C₆₇	semidirect
296. #466	C₂²×C₆₇	semidirect
297. C₄×D₈	C₂²×C₆₇	semidirect
298. C₈×D₄	C₂²×C₆₇	semidirect
299. #427	C₂²×C₆₇	semidirect
304. C₂×#107	C₂²×C₆₇	semidirect
519. C₂	C₆₇×#3339	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×#553×C₆₇	2
C₆₇×#3339	2
C₂×#499×C₆₇	2
C₂×#481×C₆₇	2
C₂×#482×C₆₇	2
C₂×#478×C₆₇	2
C₂×#466×C₆₇	2
C₂×C₄×D₈×C₆₇	2
C₂×C₈×D₄×C₆₇	2
C₂×#427×C₆₇	2
C₂²×#107×C₆₇	2
C₂×#3339	67

Subgroups — C2×C67×#3339 #30740481

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#3339 #30740481