Subgroups — C₂×C₆₇×#3364 #30740552

Order 17152.

1830 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#557×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#556×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#3364	C₂	1	16	yes	maximal, direct product factor
4	C₂×#488×C₆₇	C₂	1	2	yes	maximal
5	C₂×#487×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#479×C₆₇	C₂	1	2	yes	maximal, complemented
7	C₂×#474×C₆₇	C₂	1	1	yes	maximal, characteristic
8	C₂×#464×C₆₇	C₂	1	2	yes	maximal
9	C₂×#462×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₈×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₄×Q₁₆×C₆₇	C₂	1	2	yes	maximal, complemented
12	C₂×C₄×SD₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
13	C₂²×Q₁₆×C₆₇	C₂²	1	2	yes	—
14	C₂²×SD₁₆×C₆₇	C₂²	1	1	yes	characteristic
15	#557×C₆₇	—	2	4	no	—
16	#557×C₆₇	C₂²	1	8	yes	complemented
17	#556×C₆₇	—	2	4	no	—
18	#556×C₆₇	C₂²	1	8	yes	—
19	C₂×C₄⋊Q₈×C₆₇	—	2	1	no	—
20	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×#126×C₆₇	—	2	2	no	—
23	C₂×#125×C₆₇	—	2	2	no	—
24	C₂×#125×C₆₇	C₂²	1	2	yes	—
25	C₂×#124×C₆₇	C₂²	1	2	yes	—
26	C₂×C₂²⋊Q₈×C₆₇	—	2	1	no	—
27	C₂×C₄×Q₈×C₆₇	—	2	1	no	—
28	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
29	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
30	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
31	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
32	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
33	C₂×#117×C₆₇	—	2	1	no	—
34	#488×C₆₇	C₂²	1	16	yes	—
35	#487×C₆₇	C₂²	1	8	yes	complemented
36	#479×C₆₇	C₂²	1	16	yes	complemented
37	#474×C₆₇	C₂²	1	8	yes	—
38	#464×C₆₇	C₂²	1	16	yes	—
39	#462×C₆₇	C₂²	1	8	yes	complemented
40	C₈×Q₈×C₆₇	C₂²	1	8	yes	complemented
41	C₄×Q₁₆×C₆₇	C₂²	1	16	yes	complemented
42	C₄×SD₁₆×C₆₇	C₂²	1	8	yes	—
43	C₂×#107×C₆₇	C₂²	1	2	yes	—
44	C₂×#106×C₆₇	C₂²	1	1	yes	characteristic
45	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
46	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
47	C₂×#103×C₆₇	C₂²	1	2	yes	—
48	C₂×#103×C₆₇	C₂²	1	2	yes	—
49	C₂×#103×C₆₇	C₂²	1	2	yes	—
50	C₂×#103×C₆₇	C₂²	1	1	yes	characteristic
51	C₂×#102×C₆₇	C₂²	1	2	yes	—
52	C₂×#102×C₆₇	C₂²	1	1	yes	characteristic
53	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
54	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
55	C₂²×Q₈×C₆₇	—	2	1	no	—
56	C₂²×Q₈×C₆₇	D₄	1	1	yes	characteristic
57	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
58	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
59	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
60	C₂³×C₄×C₆₇	—	2	1	no	—
61	C₂×Q₁₆×C₆₇	—	2	8	no	—
62	C₂×Q₁₆×C₆₇	C₂³	1	8	yes	—
63	C₂×SD₁₆×C₆₇	—	2	4	no	—
64	C₂×SD₁₆×C₆₇	C₂³	1	4	yes	—
65	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
66	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
67	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
68	C₄⋊Q₈×C₆₇	—	2	8	no	—
69	C₄⋊Q₈×C₆₇	—	2	2	no	—
70	C₄⋊Q₈×C₆₇	—	2	2	no	—
71	C₄⋊Q₈×C₆₇	C₂³	1	4	yes	—
72	C₄⋊Q₈×C₆₇	C₂³	1	4	yes	—
73	#126×C₆₇	—	2	16	no	—
74	#125×C₆₇	—	2	16	no	—
75	#125×C₆₇	—	2	4	no	—
76	#125×C₆₇	C₂³	1	8	yes	—
77	#124×C₆₇	—	2	4	no	—
78	#124×C₆₇	C₂³	1	8	yes	—
79	C₂²⋊Q₈×C₆₇	—	2	8	no	—
80	C₄×Q₈×C₆₇	—	2	8	no	—
81	C₄×Q₈×C₆₇	—	2	4	no	—
82	C₄×Q₈×C₆₇	—	2	4	no	—
83	C₄×Q₈×C₆₇	—	2	2	no	—
84	C₄×Q₈×C₆₇	D₄	1	4	yes	—
85	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
86	C₄×Q₈×C₆₇	C₂³	1	8	yes	—
87	C₄×Q₈×C₆₇	C₂³	1	8	yes	—
88	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
89	C₄×D₄×C₆₇	—	2	2	no	—
90	C₄×D₄×C₆₇	C₂³	1	4	yes	—
91	#117×C₆₇	—	2	8	no	—
92	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
93	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
94	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
95	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
96	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
97	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
98	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
99	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
100	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
101	C₂×C₄⋊C₄×C₆₇	D₄	1	2	yes	—
102	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
103	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
104	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
105	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
106	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
107	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
108	C₂×C₄²×C₆₇	—	2	2	no	—
109	C₂×C₄²×C₆₇	—	2	2	no	—
110	C₂×C₄²×C₆₇	—	2	1	no	—
111	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
112	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
113	#107×C₆₇	C₂³	1	8	yes	—
114	#106×C₆₇	C₂³	1	4	yes	—
115	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
116	C₄⋊C₈×C₆₇	C₂³	1	8	yes	—
117	#103×C₆₇	C₂³	1	8	yes	—
118	#103×C₆₇	C₂³	1	8	yes	—
119	#103×C₆₇	C₂³	1	8	yes	—
120	#103×C₆₇	C₂³	1	4	yes	—
121	#102×C₆₇	C₂³	1	8	yes	—
122	#102×C₆₇	C₂³	1	4	yes	—
123	C₄×C₈×C₆₇	C₂³	1	4	yes	—
124	C₄×C₈×C₆₇	C₂³	1	8	yes	—
125	C₂⁴×C₆₇	—	2	1	no	—
126	C₂×Q₈×C₆₇	—	2	4	no	—
127	C₂×Q₈×C₆₇	—	2	1	no	—
128	C₂×Q₈×C₆₇	—	2	2	no	—
129	C₂×Q₈×C₆₇	—	2	4	no	—
130	C₂×Q₈×C₆₇	—	2	2	no	—
131	C₂×Q₈×C₆₇	—	2	4	no	—
132	C₂×Q₈×C₆₇	—	2	4	no	—
133	C₂×Q₈×C₆₇	—	4	4	no	—
134	C₂×Q₈×C₆₇	C₂×D₄	1	4	yes	—
135	C₂×Q₈×C₆₇	#41	1	4	yes	—
136	C₂×Q₈×C₆₇	C₂⁴	1	4	yes	—
137	C₂×Q₈×C₆₇	C₂⁴	1	2	yes	—
138	C₂×D₄×C₆₇	—	2	1	no	—
139	C₂×D₄×C₆₇	—	2	2	no	—
140	C₂×D₄×C₆₇	—	2	2	no	—
141	C₂×D₄×C₆₇	C₂⁴	1	2	yes	—
142	C₂²×C₄×C₆₇	—	2	1	no	—
143	C₂²×C₄×C₆₇	—	2	1	no	—
144	C₂²×C₄×C₆₇	—	2	2	no	—
145	C₂²×C₄×C₆₇	—	2	1	no	—
146	C₂²×C₄×C₆₇	—	2	2	no	—
147	C₂²×C₄×C₆₇	—	2	4	no	—
148	C₂²×C₄×C₆₇	—	4	4	no	—
149	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
150	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
151	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
152	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
153	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
154	Q₁₆×C₆₇	—	2	16	no	—
155	SD₁₆×C₆₇	—	2	8	no	—
156	C₂×C₈×C₆₇	—	2	2	no	—
157	C₂×C₈×C₆₇	—	2	4	no	—
158	C₂×C₈×C₆₇	—	2	2	no	—
159	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
160	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
161	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
162	C₄⋊C₄×C₆₇	—	2	4	no	—
163	C₄⋊C₄×C₆₇	—	2	8	no	—
164	C₄⋊C₄×C₆₇	—	2	8	no	—
165	C₄⋊C₄×C₆₇	—	2	4	no	—
166	C₄⋊C₄×C₆₇	—	2	8	no	—
167	C₄⋊C₄×C₆₇	—	2	4	no	—
168	C₄⋊C₄×C₆₇	—	2	8	no	—
169	C₄⋊C₄×C₆₇	—	2	4	no	—
170	C₄⋊C₄×C₆₇	—	2	1	no	—
171	C₄⋊C₄×C₆₇	—	2	1	no	—
172	C₄⋊C₄×C₆₇	—	2	2	no	—
173	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
174	C₄⋊C₄×C₆₇	C₂×D₄	1	8	yes	—
175	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
176	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
177	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
178	C₂²⋊C₄×C₆₇	—	2	8	no	—
179	C₂²⋊C₄×C₆₇	—	2	4	no	—
180	C₂²⋊C₄×C₆₇	—	2	8	no	—
181	C₄²×C₆₇	—	2	8	no	—
182	C₄²×C₆₇	—	2	8	no	—
183	C₄²×C₆₇	—	2	4	no	—
184	C₄²×C₆₇	C₂×D₄	1	2	yes	—
185	C₄²×C₆₇	C₂×D₄	1	4	yes	—
186	C₄²×C₆₇	C₂⁴	1	2	yes	—
187	C₄²×C₆₇	C₂⁴	1	4	yes	—
188	C₂³×C₆₇	—	2	2	no	—
189	C₂³×C₆₇	—	4	2	no	—
190	C₂³×C₆₇	—	4	2	no	—
191	C₂³×C₆₇	—	4	2	no	—
192	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
193	Q₈×C₆₇	—	2	4	no	—
194	Q₈×C₆₇	—	2	8	no	—
195	Q₈×C₆₇	—	2	2	no	—
196	Q₈×C₆₇	—	2	4	no	—
197	Q₈×C₆₇	—	2	4	no	—
198	Q₈×C₆₇	—	4	4	no	—
199	Q₈×C₆₇	—	4	8	no	—
200	Q₈×C₆₇	C₂×#41	1	4	yes	—
201	D₄×C₆₇	—	2	2	no	—
202	D₄×C₆₇	—	2	2	no	—
203	D₄×C₆₇	—	4	2	no	—
204	C₂×C₄×C₆₇	—	2	4	no	—
205	C₂×C₄×C₆₇	—	2	4	no	—
206	C₂×C₄×C₆₇	—	2	4	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	4	no	—
211	C₂×C₄×C₆₇	—	2	2	no	—
212	C₂×C₄×C₆₇	—	2	4	no	—
213	C₂×C₄×C₆₇	—	4	4	no	—
214	C₂×C₄×C₆₇	—	4	2	no	—
215	C₂×C₄×C₆₇	—	4	4	no	—
216	C₂×C₄×C₆₇	—	4	2	no	—
217	C₂×C₄×C₆₇	—	4	8	no	—
218	C₂×C₄×C₆₇	#135	1	4	yes	—
219	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
220	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
221	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
222	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
223	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
224	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
225	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
226	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
227	C₈×C₆₇	—	2	2	no	—
228	C₈×C₆₇	—	2	4	no	—
229	C₈×C₆₇	—	2	2	no	—
230	C₂²×C₆₇	—	4	2	no	—
231	C₂²×C₆₇	—	4	2	no	—
232	C₂²×C₆₇	—	4	2	no	—
233	C₂²×C₆₇	—	8	4	no	—
234	C₂²×C₆₇	#548	1	2	yes	—
235	C₂²×C₆₇	C₂×#135	1	2	yes	—
236	C₂²×C₆₇	#578	1	2	yes	—
237	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
238	C₄×C₆₇	—	2	4	no	—
239	C₄×C₆₇	—	2	4	no	—
240	C₄×C₆₇	—	2	4	no	—
241	C₄×C₆₇	—	2	2	no	—
242	C₄×C₆₇	—	4	4	no	—
243	C₄×C₆₇	—	4	2	no	—
244	C₄×C₆₇	—	4	4	no	—
245	C₄×C₆₇	—	4	2	no	—
246	C₄×C₆₇	#548	1	2	yes	—
247	C₄×C₆₇	C₂×#135	1	4	yes	—
248	C₄×C₆₇	C₂²×#41	1	1	yes	characteristic
249	C₄×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
250	C₂×#3364	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
251	C₂×C₆₇	—	8	2	no	—
252	C₂×C₆₇	#3364	1	4	yes	direct product factor
253	C₂×C₆₇	C₂×#548	1	1	yes	characteristic
254	C₂×C₆₇	C₂²×#135	1	1	yes	characteristic
255	C₂×C₆₇	C₂×#578	1	1	yes	characteristic
256	C₂×#557	C₂×C₆₇	1	1	yes	characteristic, complemented
257	C₂×#556	C₂×C₆₇	1	1	yes	characteristic
258	#3364	C₂×C₆₇	1	16	yes	direct product factor
259	C₂×#488	C₂×C₆₇	1	2	yes	—
260	C₂×#487	C₂×C₆₇	1	1	yes	characteristic, complemented
261	C₂×#479	C₂×C₆₇	1	2	yes	complemented
262	C₂×#474	C₂×C₆₇	1	1	yes	characteristic
263	C₂×#464	C₂×C₆₇	1	2	yes	—
264	C₂×#462	C₂×C₆₇	1	1	yes	characteristic, complemented
265	C₂×C₈×Q₈	C₂×C₆₇	1	1	yes	characteristic, complemented
266	C₂×C₄×Q₁₆	C₂×C₆₇	1	2	yes	complemented
267	C₂×C₄×SD₁₆	C₂×C₆₇	1	1	yes	characteristic
268	C₆₇	C₂×#3364	1	1	yes	characteristic, direct product factor, Sylow
269	C₂²×Q₁₆	C₂²×C₆₇	1	2	yes	—
270	C₂²×SD₁₆	C₂²×C₆₇	1	1	yes	characteristic
271	#557	—	2	4	no	—
272	#557	C₂²×C₆₇	1	8	yes	complemented
273	#556	—	2	4	no	—
274	#556	C₂²×C₆₇	1	8	yes	—
275	C₂×C₄⋊Q₈	—	2	1	no	—
276	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
277	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
278	C₂×#126	—	2	2	no	—
279	C₂×#125	—	2	2	no	—
280	C₂×#125	C₂²×C₆₇	1	2	yes	—
281	C₂×#124	C₂²×C₆₇	1	2	yes	—
282	C₂×C₂²⋊Q₈	—	2	1	no	—
283	C₂×C₄×Q₈	—	2	1	no	—
284	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
285	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
286	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
287	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
288	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
289	C₂×#117	—	2	1	no	—
290	#488	C₂²×C₆₇	1	16	yes	—
291	#487	C₂²×C₆₇	1	8	yes	complemented
292	#479	C₂²×C₆₇	1	16	yes	complemented
293	#474	C₂²×C₆₇	1	8	yes	—
294	#464	C₂²×C₆₇	1	16	yes	—
295	#462	C₂²×C₆₇	1	8	yes	complemented
296	C₈×Q₈	C₂²×C₆₇	1	8	yes	complemented
297	C₄×Q₁₆	C₂²×C₆₇	1	16	yes	complemented
298	C₄×SD₁₆	C₂²×C₆₇	1	8	yes	—
299	C₂×#107	C₂²×C₆₇	1	2	yes	—
300	C₂×#106	C₂²×C₆₇	1	1	yes	characteristic
301	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
302	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
303	C₂×#103	C₂²×C₆₇	1	2	yes	—
304	C₂×#103	C₂²×C₆₇	1	2	yes	—
305	C₂×#103	C₂²×C₆₇	1	2	yes	—
306	C₂×#103	C₂²×C₆₇	1	1	yes	characteristic
307	C₂×#102	C₂²×C₆₇	1	1	yes	characteristic
308	C₂×#102	C₂²×C₆₇	1	2	yes	—
309	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
310	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
311	C₂²×Q₈	—	2	1	no	—
312	C₂²×Q₈	D₄×C₆₇	1	1	yes	characteristic
313	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
314	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
315	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
316	C₂³×C₄	—	2	1	no	—
317	C₂×Q₁₆	—	2	8	no	—
318	C₂×Q₁₆	C₂³×C₆₇	1	8	yes	—
319	C₂×SD₁₆	—	2	4	no	—
320	C₂×SD₁₆	C₂³×C₆₇	1	4	yes	—
321	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
322	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
323	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
324	C₄⋊Q₈	—	2	8	no	—
325	C₄⋊Q₈	—	2	2	no	—
326	C₄⋊Q₈	—	2	2	no	—
327	C₄⋊Q₈	C₂³×C₆₇	1	4	yes	—
328	C₄⋊Q₈	C₂³×C₆₇	1	4	yes	—
329	#126	—	2	16	no	—
330	#125	—	2	16	no	—
331	#125	—	2	4	no	—
332	#125	C₂³×C₆₇	1	8	yes	—
333	#124	—	2	4	no	—
334	#124	C₂³×C₆₇	1	8	yes	—
335	C₂²⋊Q₈	—	2	8	no	—
336	C₄×Q₈	—	2	8	no	—
337	C₄×Q₈	—	2	4	no	—
338	C₄×Q₈	—	2	4	no	—
339	C₄×Q₈	—	2	2	no	—
340	C₄×Q₈	D₄×C₆₇	1	4	yes	—
341	C₄×Q₈	C₂³×C₆₇	1	8	yes	—
342	C₄×Q₈	C₂³×C₆₇	1	4	yes	—
343	C₄×Q₈	C₂³×C₆₇	1	8	yes	—
344	C₄×Q₈	C₂³×C₆₇	1	4	yes	—
345	C₄×D₄	—	2	2	no	—
346	C₄×D₄	C₂³×C₆₇	1	4	yes	—
347	#117	—	2	8	no	—
348	C₂×C₄⋊C₄	—	2	1	no	—
349	C₂×C₄⋊C₄	—	2	2	no	—
350	C₂×C₄⋊C₄	—	2	2	no	—
351	C₂×C₄⋊C₄	—	2	1	no	—
352	C₂×C₄⋊C₄	—	2	2	no	—
353	C₂×C₄⋊C₄	—	2	1	no	—
354	C₂×C₄⋊C₄	—	2	2	no	—
355	C₂×C₄⋊C₄	—	2	1	no	—
356	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
357	C₂×C₄⋊C₄	D₄×C₆₇	1	2	yes	—
358	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
359	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
360	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
361	C₂×C₂²⋊C₄	—	2	2	no	—
362	C₂×C₂²⋊C₄	—	2	1	no	—
363	C₂×C₂²⋊C₄	—	2	2	no	—
364	C₂×C₄²	—	2	2	no	—
365	C₂×C₄²	—	2	2	no	—
366	C₂×C₄²	—	2	1	no	—
367	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
368	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
369	#107	C₂³×C₆₇	1	8	yes	—
370	#106	C₂³×C₆₇	1	4	yes	—
371	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
372	C₄⋊C₈	C₂³×C₆₇	1	8	yes	—
373	#103	C₂³×C₆₇	1	4	yes	—
374	#103	C₂³×C₆₇	1	8	yes	—
375	#103	C₂³×C₆₇	1	8	yes	—
376	#103	C₂³×C₆₇	1	8	yes	—
377	#102	C₂³×C₆₇	1	4	yes	—
378	#102	C₂³×C₆₇	1	8	yes	—
379	C₄×C₈	C₂³×C₆₇	1	4	yes	—
380	C₄×C₈	C₂³×C₆₇	1	8	yes	—
381	C₂⁴	—	2	1	no	—
382	C₂×Q₈	—	2	4	no	—
383	C₂×Q₈	—	2	1	no	—
384	C₂×Q₈	—	2	2	no	—
385	C₂×Q₈	—	2	4	no	—
386	C₂×Q₈	—	2	2	no	—
387	C₂×Q₈	—	2	4	no	—
388	C₂×Q₈	—	2	4	no	—
389	C₂×Q₈	—	4	4	no	—
390	C₂×Q₈	C₂×D₄×C₆₇	1	4	yes	—
391	C₂×Q₈	#41×C₆₇	1	4	yes	—
392	C₂×Q₈	C₂⁴×C₆₇	1	4	yes	—
393	C₂×Q₈	C₂⁴×C₆₇	1	2	yes	—
394	C₂×D₄	—	2	1	no	—
395	C₂×D₄	—	2	2	no	—
396	C₂×D₄	—	2	2	no	—
397	C₂×D₄	C₂⁴×C₆₇	1	2	yes	—
398	C₂²×C₄	—	2	1	no	—
399	C₂²×C₄	—	2	1	no	—
400	C₂²×C₄	—	2	2	no	—
401	C₂²×C₄	—	2	1	no	—
402	C₂²×C₄	—	2	2	no	—
403	C₂²×C₄	—	2	4	no	—
404	C₂²×C₄	—	4	4	no	—
405	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
406	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
407	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
408	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
409	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
410	Q₁₆	—	2	16	no	—
411	SD₁₆	—	2	8	no	—
412	C₂×C₈	—	2	2	no	—
413	C₂×C₈	—	2	4	no	—
414	C₂×C₈	—	2	2	no	—
415	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
416	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
417	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
418	C₄⋊C₄	—	2	4	no	—
419	C₄⋊C₄	—	2	8	no	—
420	C₄⋊C₄	—	2	8	no	—
421	C₄⋊C₄	—	2	4	no	—
422	C₄⋊C₄	—	2	8	no	—
423	C₄⋊C₄	—	2	4	no	—
424	C₄⋊C₄	—	2	8	no	—
425	C₄⋊C₄	—	2	4	no	—
426	C₄⋊C₄	—	2	1	no	—
427	C₄⋊C₄	—	2	1	no	—
428	C₄⋊C₄	—	2	2	no	—
429	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
430	C₄⋊C₄	C₂×D₄×C₆₇	1	8	yes	—
431	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
432	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
433	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
434	C₂²⋊C₄	—	2	8	no	—
435	C₂²⋊C₄	—	2	4	no	—
436	C₂²⋊C₄	—	2	8	no	—
437	C₄²	—	2	8	no	—
438	C₄²	—	2	8	no	—
439	C₄²	—	2	4	no	—
440	C₄²	C₂×D₄×C₆₇	1	2	yes	—
441	C₄²	C₂×D₄×C₆₇	1	4	yes	—
442	C₄²	C₂⁴×C₆₇	1	2	yes	—
443	C₄²	C₂⁴×C₆₇	1	4	yes	—
444	C₂³	—	2	2	no	—
445	C₂³	—	4	2	no	—
446	C₂³	—	4	2	no	—
447	C₂³	—	4	2	no	—
448	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
449	Q₈	—	2	4	no	—
450	Q₈	—	2	8	no	—
451	Q₈	—	2	2	no	—
452	Q₈	—	2	4	no	—
453	Q₈	—	2	4	no	—
454	Q₈	—	4	4	no	—
455	Q₈	—	4	8	no	—
456	Q₈	C₂×#41×C₆₇	1	4	yes	—
457	D₄	—	2	2	no	—
458	D₄	—	2	2	no	—
459	D₄	—	4	2	no	—
460	C₂×C₄	—	2	4	no	—
461	C₂×C₄	—	2	4	no	—
462	C₂×C₄	—	2	4	no	—
463	C₂×C₄	—	2	2	no	—
464	C₂×C₄	—	2	2	no	—
465	C₂×C₄	—	2	2	no	—
466	C₂×C₄	—	2	4	no	—
467	C₂×C₄	—	2	2	no	—
468	C₂×C₄	—	2	4	no	—
469	C₂×C₄	—	4	4	no	—
470	C₂×C₄	—	4	2	no	—
471	C₂×C₄	—	4	4	no	—
472	C₂×C₄	—	4	2	no	—
473	C₂×C₄	—	4	8	no	—
474	C₂×C₄	#135×C₆₇	1	4	yes	—
475	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
476	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
477	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
478	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
479	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
480	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
481	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
482	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
483	C₈	—	2	2	no	—
484	C₈	—	2	4	no	—
485	C₈	—	2	2	no	—
486	C₂²	—	4	2	no	—
487	C₂²	—	4	2	no	—
488	C₂²	—	4	2	no	—
489	C₂²	—	8	4	no	—
490	C₂²	#548×C₆₇	1	2	yes	—
491	C₂²	C₂×#135×C₆₇	1	2	yes	—
492	C₂²	#578×C₆₇	1	2	yes	—
493	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
494	C₄	—	2	4	no	—
495	C₄	—	2	4	no	—
496	C₄	—	2	2	no	—
497	C₄	—	2	4	no	—
498	C₄	—	4	2	no	—
499	C₄	—	4	4	no	—
500	C₄	—	4	2	no	—
501	C₄	—	4	4	no	—
502	C₄	#548×C₆₇	1	2	yes	—
503	C₄	C₂×#135×C₆₇	1	4	yes	—
504	C₄	C₂²×#41×C₆₇	1	1	yes	characteristic
505	C₄	C₂×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
506	C₂	—	8	2	no	—
507	C₂	C₆₇×#3364	1	4	yes	direct product factor
508	C₂	C₂×#548×C₆₇	1	1	yes	characteristic
509	C₂	C₂²×#135×C₆₇	1	1	yes	characteristic
510	C₂	C₂×#578×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
prime radical	C₂²×D₄×C₆₇	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂×C₄×Q₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#557×C₆₇	C₂	semidirect
3. C₆₇×#3364	C₂	direct (N × H)
5. C₂×#487×C₆₇	C₂	semidirect
6. C₂×#479×C₆₇	C₂	semidirect
9. C₂×#462×C₆₇	C₂	semidirect
10. C₂×C₈×Q₈×C₆₇	C₂	semidirect
11. C₂×C₄×Q₁₆×C₆₇	C₂	semidirect
16. #557×C₆₇	C₂²	semidirect
35. #487×C₆₇	C₂²	semidirect
36. #479×C₆₇	C₂²	semidirect
39. #462×C₆₇	C₂²	semidirect
40. C₈×Q₈×C₆₇	C₂²	semidirect
41. C₄×Q₁₆×C₆₇	C₂²	semidirect
250. C₂×#3364	C₆₇	direct (N × H)
252. C₂×C₆₇	#3364	direct (N × H)
256. C₂×#557	C₂×C₆₇	semidirect
258. #3364	C₂×C₆₇	direct (N × H)
260. C₂×#487	C₂×C₆₇	semidirect
261. C₂×#479	C₂×C₆₇	semidirect
264. C₂×#462	C₂×C₆₇	semidirect
265. C₂×C₈×Q₈	C₂×C₆₇	semidirect
266. C₂×C₄×Q₁₆	C₂×C₆₇	semidirect
268. C₆₇	C₂×#3364	direct (N × H)
272. #557	C₂²×C₆₇	semidirect
291. #487	C₂²×C₆₇	semidirect
292. #479	C₂²×C₆₇	semidirect
295. #462	C₂²×C₆₇	semidirect
296. C₈×Q₈	C₂²×C₆₇	semidirect
297. C₄×Q₁₆	C₂²×C₆₇	semidirect
507. C₂	C₆₇×#3364	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×#557×C₆₇	2
C₂×#556×C₆₇	2
C₆₇×#3364	2
C₂×#488×C₆₇	2
C₂×#487×C₆₇	2
C₂×#479×C₆₇	2
C₂×#474×C₆₇	2
C₂×#464×C₆₇	2
C₂×#462×C₆₇	2
C₂×C₈×Q₈×C₆₇	2
C₂×C₄×Q₁₆×C₆₇	2
C₂×C₄×SD₁₆×C₆₇	2
C₂×#3364	67

Subgroups — C2×C67×#3364 #30740552

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#3364 #30740552