Subgroups — C₆₇×#33178 #30712429

Order 17152.

1126 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#492×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#468×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₂×#465×C₆₇	C₂	1	2	yes	maximal, complemented
4	C₂×#405×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#2621	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#2619	C₂	1	1	yes	maximal, characteristic
7	C₆₇×#1973	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1968	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1961	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#1960	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#1846	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×SD₁₆×C₆₇	C₂²	1	2	yes	—
13	C₂²×SD₁₆×C₆₇	C₂²	1	2	yes	—
14	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
15	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
19	C₂×#121×C₆₇	C₂²	1	2	yes	—
20	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
21	#492×C₆₇	—	2	4	no	—
22	#468×C₆₇	—	2	8	no	—
23	#465×C₆₇	—	2	8	no	—
24	C₂×#107×C₆₇	C₂²	1	2	yes	—
25	C₂×#106×C₆₇	C₂²	1	2	yes	—
26	C₂×#103×C₆₇	C₂²	1	2	yes	—
27	C₂×#103×C₆₇	C₂²	1	2	yes	—
28	C₂×#102×C₆₇	C₂²	1	2	yes	—
29	C₂×#102×C₆₇	C₂²	1	2	yes	—
30	#405×C₆₇	—	2	4	no	—
31	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
32	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
33	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
34	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
35	#398×C₆₇	—	2	2	no	—
36	#397×C₆₇	—	2	2	no	—
37	#394×C₆₇	—	2	2	no	—
38	#392×C₆₇	—	2	2	no	—
39	#391×C₆₇	C₂²	1	2	yes	—
40	#390×C₆₇	C₂²	1	2	yes	—
41	#382×C₆₇	C₂²	1	1	yes	characteristic
42	#382×C₆₇	C₂²	1	1	yes	characteristic
43	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
44	C₂²×D₄×C₆₇	—	2	2	no	—
45	C₂²×D₄×C₆₇	—	2	2	no	—
46	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
47	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
48	C₂×SD₁₆×C₆₇	—	2	4	no	—
49	C₂×SD₁₆×C₆₇	—	2	4	no	—
50	C₂×SD₁₆×C₆₇	—	2	4	no	—
51	C₂×SD₁₆×C₆₇	—	2	4	no	—
52	C₂×SD₁₆×C₆₇	—	2	4	no	—
53	C₂×SD₁₆×C₆₇	—	2	4	no	—
54	C₂×M₁₆×C₆₇	—	2	2	no	—
55	C₂×M₁₆×C₆₇	—	2	2	no	—
56	C₂×M₁₆×C₆₇	D₄	1	4	yes	complemented
57	C₂²×C₈×C₆₇	—	2	2	no	—
58	C₂²×C₈×C₆₇	—	2	2	no	—
59	C₂²×C₈×C₆₇	D₄	1	4	yes	complemented
60	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
61	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
62	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
63	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
64	C₄⋊Q₈×C₆₇	—	2	2	no	—
65	C₄⋊Q₈×C₆₇	—	2	2	no	—
66	#127×C₆₇	—	2	2	no	—
67	#127×C₆₇	—	2	2	no	—
68	C₂²⋊Q₈×C₆₇	—	2	4	no	—
69	C₂²⋊Q₈×C₆₇	—	2	4	no	—
70	#121×C₆₇	—	2	4	no	—
71	#121×C₆₇	—	2	4	no	—
72	#117×C₆₇	—	2	2	no	—
73	#117×C₆₇	—	2	2	no	—
74	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
75	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
76	C₂×C₄⋊C₄×C₆₇	D₄	1	2	yes	—
77	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
78	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
79	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
80	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
81	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
82	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
83	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
84	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
85	#107×C₆₇	—	2	4	no	—
86	#106×C₆₇	—	2	4	no	—
87	#103×C₆₇	—	2	4	no	—
88	#103×C₆₇	—	2	4	no	—
89	#102×C₆₇	—	2	4	no	—
90	#102×C₆₇	—	2	4	no	—
91	#97×C₆₇	—	2	2	no	—
92	#97×C₆₇	—	2	2	no	—
93	C₄×C₈×C₆₇	—	2	2	no	—
94	C₄×C₈×C₆₇	D₄	1	4	yes	complemented
95	#95×C₆₇	—	2	2	no	—
96	#95×C₆₇	—	2	2	no	—
97	#95×C₆₇	—	2	4	no	—
98	C₂⁴×C₆₇	—	2	2	no	—
99	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
100	C₂×Q₈×C₆₇	—	2	2	no	—
101	C₂×Q₈×C₆₇	—	2	2	no	—
102	C₂×Q₈×C₆₇	—	2	2	no	—
103	C₂×Q₈×C₆₇	—	2	2	no	—
104	C₂×Q₈×C₆₇	—	2	2	no	—
105	C₂×Q₈×C₆₇	—	2	2	no	—
106	C₂×D₄×C₆₇	—	2	2	no	—
107	C₂×D₄×C₆₇	—	2	2	no	—
108	C₂×D₄×C₆₇	—	2	2	no	—
109	C₂×D₄×C₆₇	—	2	2	no	—
110	C₂×D₄×C₆₇	—	2	2	no	—
111	C₂×D₄×C₆₇	—	2	2	no	—
112	C₂×D₄×C₆₇	—	4	4	no	—
113	C₂×D₄×C₆₇	—	4	4	no	—
114	C₂×D₄×C₆₇	—	8	4	no	—
115	C₂×D₄×C₆₇	—	8	4	no	—
116	C₂²×C₄×C₆₇	—	2	1	no	—
117	C₂²×C₄×C₆₇	—	2	1	no	—
118	C₂²×C₄×C₆₇	—	2	1	no	—
119	C₂²×C₄×C₆₇	—	2	1	no	—
120	C₂²×C₄×C₆₇	—	2	2	no	—
121	C₂²×C₄×C₆₇	—	2	2	no	—
122	C₂²×C₄×C₆₇	—	2	2	no	—
123	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
124	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
125	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
126	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
127	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
128	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
129	SD₁₆×C₆₇	—	4	8	no	—
130	SD₁₆×C₆₇	—	4	8	no	—
131	M₁₆×C₆₇	—	2	8	no	—
132	C₂×C₈×C₆₇	—	2	1	no	—
133	C₂×C₈×C₆₇	—	2	1	no	—
134	C₂×C₈×C₆₇	—	2	1	no	—
135	C₂×C₈×C₆₇	—	2	1	no	—
136	C₂×C₈×C₆₇	—	2	8	no	—
137	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
138	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
139	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
140	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
141	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
142	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
143	C₂×C₈×C₆₇	#41	1	2	yes	—
144	C₂×C₈×C₆₇	#41	1	2	yes	—
145	C₄⋊C₄×C₆₇	—	2	4	no	—
146	C₄⋊C₄×C₆₇	—	2	2	no	—
147	C₄⋊C₄×C₆₇	—	2	2	no	—
148	C₄⋊C₄×C₆₇	—	2	2	no	—
149	C₄⋊C₄×C₆₇	—	2	2	no	—
150	C₄⋊C₄×C₆₇	—	4	4	no	—
151	C₄⋊C₄×C₆₇	—	4	4	no	—
152	C₂²⋊C₄×C₆₇	—	2	4	no	—
153	C₂²⋊C₄×C₆₇	—	4	4	no	—
154	C₂²⋊C₄×C₆₇	—	4	4	no	—
155	C₂²⋊C₄×C₆₇	—	8	4	no	—
156	C₄²×C₆₇	—	2	1	no	—
157	C₄²×C₆₇	—	2	1	no	—
158	C₄²×C₆₇	C₂×D₄	1	2	yes	—
159	C₄²×C₆₇	#41	1	2	yes	—
160	C₂³×C₆₇	—	2	1	no	—
161	C₂³×C₆₇	—	2	1	no	—
162	C₂³×C₆₇	—	2	4	no	—
163	C₂³×C₆₇	—	4	2	no	—
164	C₂³×C₆₇	—	4	2	no	—
165	C₂³×C₆₇	—	4	2	no	—
166	C₂³×C₆₇	—	4	4	no	—
167	C₂³×C₆₇	—	4	4	no	—
168	C₂³×C₆₇	#127	1	2	yes	—
169	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
170	Q₈×C₆₇	—	4	2	no	—
171	Q₈×C₆₇	—	4	2	no	—
172	Q₈×C₆₇	—	4	2	no	—
173	Q₈×C₆₇	—	4	2	no	—
174	D₄×C₆₇	—	4	2	no	—
175	D₄×C₆₇	—	4	2	no	—
176	D₄×C₆₇	—	4	2	no	—
177	D₄×C₆₇	—	4	2	no	—
178	D₄×C₆₇	—	8	8	no	—
179	D₄×C₆₇	—	8	8	no	—
180	C₂×C₄×C₆₇	—	2	1	no	—
181	C₂×C₄×C₆₇	—	2	1	no	—
182	C₂×C₄×C₆₇	—	2	4	no	—
183	C₂×C₄×C₆₇	—	2	2	no	—
184	C₂×C₄×C₆₇	—	2	2	no	—
185	C₂×C₄×C₆₇	—	2	2	no	—
186	C₂×C₄×C₆₇	—	2	2	no	—
187	C₂×C₄×C₆₇	—	2	2	no	—
188	C₂×C₄×C₆₇	—	2	4	no	—
189	C₂×C₄×C₆₇	—	4	2	no	—
190	C₂×C₄×C₆₇	—	4	2	no	—
191	C₂×C₄×C₆₇	—	4	2	no	—
192	C₂×C₄×C₆₇	—	4	2	no	—
193	C₂×C₄×C₆₇	—	4	2	no	—
194	C₂×C₄×C₆₇	—	4	4	no	—
195	C₂×C₄×C₆₇	—	4	4	no	—
196	C₂×C₄×C₆₇	#121	1	4	yes	—
197	C₂×C₄×C₆₇	#127	1	2	yes	—
198	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
199	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
200	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
201	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
202	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
203	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
204	C₈×C₆₇	—	2	2	no	—
205	C₈×C₆₇	—	2	2	no	—
206	C₈×C₆₇	#121	1	4	yes	—
207	C₈×C₆₇	#121	1	4	yes	—
208	C₂²×C₆₇	—	2	2	no	—
209	C₂²×C₆₇	—	2	4	no	—
210	C₂²×C₆₇	—	4	2	no	—
211	C₂²×C₆₇	—	4	2	no	—
212	C₂²×C₆₇	—	8	2	no	—
213	C₂²×C₆₇	—	8	2	no	—
214	C₂²×C₆₇	—	8	2	no	—
215	C₂²×C₆₇	—	8	4	no	—
216	C₂²×C₆₇	—	8	4	no	—
217	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
218	C₂²×C₆₇	C₂×#127	1	1	yes	characteristic
219	C₂²×C₆₇	#532	1	1	yes	characteristic
220	C₂²×C₆₇	#577	1	4	yes	—
221	C₄×C₆₇	—	2	2	no	—
222	C₄×C₆₇	—	2	4	no	—
223	C₄×C₆₇	—	4	2	no	—
224	C₄×C₆₇	—	8	2	no	—
225	C₄×C₆₇	—	8	2	no	—
226	C₄×C₆₇	—	8	2	no	—
227	C₄×C₆₇	C₂×#121	1	1	yes	characteristic
228	C₄×C₆₇	C₂×#121	1	1	yes	characteristic
229	C₄×C₆₇	#532	1	1	yes	characteristic
230	C₄×C₆₇	#532	1	1	yes	characteristic
231	#33178	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
232	C₂×C₆₇	—	4	2	no	—
233	C₂×C₆₇	—	16	2	no	—
234	C₂×C₆₇	#2856	1	1	yes	characteristic
235	C₂×C₆₇	#3057	1	2	yes	—
236	C₂×C₆₇	#3154	1	2	yes	—
237	C₂×C₆₇	#3174	1	2	yes	—
238	C₂×#492	C₂×C₆₇	1	1	yes	characteristic, complemented
239	C₂×#468	C₂×C₆₇	1	2	yes	complemented
240	C₂×#465	C₂×C₆₇	1	2	yes	complemented
241	C₂×#405	C₂×C₆₇	1	1	yes	characteristic, complemented
242	#2621	C₂×C₆₇	1	1	yes	characteristic, complemented
243	#2619	C₂×C₆₇	1	1	yes	characteristic
244	#1973	C₂×C₆₇	1	2	yes	complemented
245	#1968	C₂×C₆₇	1	2	yes	complemented
246	#1961	C₂×C₆₇	1	1	yes	characteristic, complemented
247	#1960	C₂×C₆₇	1	1	yes	characteristic, complemented
248	#1846	C₂×C₆₇	1	1	yes	characteristic, complemented
249	C₆₇	#33178	1	1	yes	characteristic, direct product factor, Sylow
250	C₂²×SD₁₆	C₂²×C₆₇	1	2	yes	—
251	C₂²×SD₁₆	C₂²×C₆₇	1	2	yes	—
252	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
253	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
254	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
255	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
256	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
257	C₂×#121	C₂²×C₆₇	1	2	yes	—
258	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
259	#492	—	2	4	no	—
260	#468	—	2	8	no	—
261	#465	—	2	8	no	—
262	C₂×#107	C₂²×C₆₇	1	2	yes	—
263	C₂×#106	C₂²×C₆₇	1	2	yes	—
264	C₂×#103	C₂²×C₆₇	1	2	yes	—
265	C₂×#103	C₂²×C₆₇	1	2	yes	—
266	C₂×#102	C₂²×C₆₇	1	2	yes	—
267	C₂×#102	C₂²×C₆₇	1	2	yes	—
268	#405	—	2	4	no	—
269	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
270	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
271	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
272	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
273	#398	—	2	2	no	—
274	#397	—	2	2	no	—
275	#394	—	2	2	no	—
276	#392	—	2	2	no	—
277	#391	C₂²×C₆₇	1	2	yes	—
278	#390	C₂²×C₆₇	1	2	yes	—
279	#382	C₂²×C₆₇	1	1	yes	characteristic
280	#382	C₂²×C₆₇	1	1	yes	characteristic
281	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
282	C₂²×D₄	—	2	2	no	—
283	C₂²×D₄	—	2	2	no	—
284	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
285	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
286	C₂×SD₁₆	—	2	4	no	—
287	C₂×SD₁₆	—	2	4	no	—
288	C₂×SD₁₆	—	2	4	no	—
289	C₂×SD₁₆	—	2	4	no	—
290	C₂×SD₁₆	—	2	4	no	—
291	C₂×SD₁₆	—	2	4	no	—
292	C₂×M₁₆	—	2	2	no	—
293	C₂×M₁₆	—	2	2	no	—
294	C₂×M₁₆	D₄×C₆₇	1	4	yes	complemented
295	C₂²×C₈	—	2	2	no	—
296	C₂²×C₈	—	2	2	no	—
297	C₂²×C₈	D₄×C₆₇	1	4	yes	complemented
298	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
299	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
300	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
301	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
302	C₄⋊Q₈	—	2	2	no	—
303	C₄⋊Q₈	—	2	2	no	—
304	#127	—	2	2	no	—
305	#127	—	2	2	no	—
306	C₂²⋊Q₈	—	2	4	no	—
307	C₂²⋊Q₈	—	2	4	no	—
308	#121	—	2	4	no	—
309	#121	—	2	4	no	—
310	#117	—	2	2	no	—
311	#117	—	2	2	no	—
312	C₂×C₄⋊C₄	—	2	2	no	—
313	C₂×C₄⋊C₄	—	2	2	no	—
314	C₂×C₄⋊C₄	D₄×C₆₇	1	2	yes	—
315	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
316	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
317	C₂×C₂²⋊C₄	—	2	2	no	—
318	C₂×C₂²⋊C₄	—	2	2	no	—
319	C₂×C₂²⋊C₄	—	2	4	no	—
320	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
321	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
322	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
323	#107	—	2	4	no	—
324	#106	—	2	4	no	—
325	#103	—	2	4	no	—
326	#103	—	2	4	no	—
327	#102	—	2	4	no	—
328	#102	—	2	4	no	—
329	#97	—	2	2	no	—
330	#97	—	2	2	no	—
331	C₄×C₈	—	2	2	no	—
332	C₄×C₈	D₄×C₆₇	1	4	yes	complemented
333	#95	—	2	2	no	—
334	#95	—	2	2	no	—
335	#95	—	2	4	no	—
336	C₂⁴	—	2	2	no	—
337	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
338	C₂×Q₈	—	2	2	no	—
339	C₂×Q₈	—	2	2	no	—
340	C₂×Q₈	—	2	2	no	—
341	C₂×Q₈	—	2	2	no	—
342	C₂×Q₈	—	2	2	no	—
343	C₂×Q₈	—	2	2	no	—
344	C₂×D₄	—	2	2	no	—
345	C₂×D₄	—	2	2	no	—
346	C₂×D₄	—	2	2	no	—
347	C₂×D₄	—	2	2	no	—
348	C₂×D₄	—	2	2	no	—
349	C₂×D₄	—	2	2	no	—
350	C₂×D₄	—	4	4	no	—
351	C₂×D₄	—	4	4	no	—
352	C₂×D₄	—	8	4	no	—
353	C₂×D₄	—	8	4	no	—
354	C₂²×C₄	—	2	1	no	—
355	C₂²×C₄	—	2	1	no	—
356	C₂²×C₄	—	2	1	no	—
357	C₂²×C₄	—	2	1	no	—
358	C₂²×C₄	—	2	2	no	—
359	C₂²×C₄	—	2	2	no	—
360	C₂²×C₄	—	2	2	no	—
361	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
362	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
363	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
364	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
365	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
366	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
367	SD₁₆	—	4	8	no	—
368	SD₁₆	—	4	8	no	—
369	M₁₆	—	2	8	no	—
370	C₂×C₈	—	2	1	no	—
371	C₂×C₈	—	2	1	no	—
372	C₂×C₈	—	2	1	no	—
373	C₂×C₈	—	2	1	no	—
374	C₂×C₈	—	2	8	no	—
375	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
376	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
377	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
378	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
379	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
380	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
381	C₂×C₈	#41×C₆₇	1	2	yes	—
382	C₂×C₈	#41×C₆₇	1	2	yes	—
383	C₄⋊C₄	—	2	4	no	—
384	C₄⋊C₄	—	2	2	no	—
385	C₄⋊C₄	—	2	2	no	—
386	C₄⋊C₄	—	2	2	no	—
387	C₄⋊C₄	—	2	2	no	—
388	C₄⋊C₄	—	4	4	no	—
389	C₄⋊C₄	—	4	4	no	—
390	C₂²⋊C₄	—	2	4	no	—
391	C₂²⋊C₄	—	4	4	no	—
392	C₂²⋊C₄	—	4	4	no	—
393	C₂²⋊C₄	—	8	4	no	—
394	C₄²	—	2	1	no	—
395	C₄²	—	2	1	no	—
396	C₄²	C₂×D₄×C₆₇	1	2	yes	—
397	C₄²	#41×C₆₇	1	2	yes	—
398	C₂³	—	2	1	no	—
399	C₂³	—	2	1	no	—
400	C₂³	—	2	4	no	—
401	C₂³	—	4	2	no	—
402	C₂³	—	4	2	no	—
403	C₂³	—	4	2	no	—
404	C₂³	—	4	4	no	—
405	C₂³	—	4	4	no	—
406	C₂³	#127×C₆₇	1	2	yes	—
407	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
408	Q₈	—	4	2	no	—
409	Q₈	—	4	2	no	—
410	Q₈	—	4	2	no	—
411	Q₈	—	4	2	no	—
412	D₄	—	4	2	no	—
413	D₄	—	4	2	no	—
414	D₄	—	4	2	no	—
415	D₄	—	4	2	no	—
416	D₄	—	8	8	no	—
417	D₄	—	8	8	no	—
418	C₂×C₄	—	2	1	no	—
419	C₂×C₄	—	2	1	no	—
420	C₂×C₄	—	2	4	no	—
421	C₂×C₄	—	2	2	no	—
422	C₂×C₄	—	2	2	no	—
423	C₂×C₄	—	2	2	no	—
424	C₂×C₄	—	2	2	no	—
425	C₂×C₄	—	2	2	no	—
426	C₂×C₄	—	2	4	no	—
427	C₂×C₄	—	4	2	no	—
428	C₂×C₄	—	4	2	no	—
429	C₂×C₄	—	4	2	no	—
430	C₂×C₄	—	4	2	no	—
431	C₂×C₄	—	4	2	no	—
432	C₂×C₄	—	4	4	no	—
433	C₂×C₄	—	4	4	no	—
434	C₂×C₄	#121×C₆₇	1	4	yes	—
435	C₂×C₄	#127×C₆₇	1	2	yes	—
436	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
437	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
438	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
439	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
440	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
441	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
442	C₈	—	2	2	no	—
443	C₈	—	2	2	no	—
444	C₈	#121×C₆₇	1	4	yes	—
445	C₈	#121×C₆₇	1	4	yes	—
446	C₂²	—	2	2	no	—
447	C₂²	—	2	4	no	—
448	C₂²	—	4	2	no	—
449	C₂²	—	4	2	no	—
450	C₂²	—	8	4	no	—
451	C₂²	—	8	2	no	—
452	C₂²	—	8	2	no	—
453	C₂²	—	8	2	no	—
454	C₂²	—	8	4	no	—
455	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
456	C₂²	C₂×#127×C₆₇	1	1	yes	characteristic
457	C₂²	#532×C₆₇	1	1	yes	characteristic
458	C₂²	#577×C₆₇	1	4	yes	—
459	C₄	—	2	4	no	—
460	C₄	—	2	2	no	—
461	C₄	—	4	2	no	—
462	C₄	—	8	2	no	—
463	C₄	—	8	2	no	—
464	C₄	—	8	2	no	—
465	C₄	C₂×#121×C₆₇	1	1	yes	characteristic
466	C₄	C₂×#121×C₆₇	1	1	yes	characteristic
467	C₄	#532×C₆₇	1	1	yes	characteristic
468	C₄	#532×C₆₇	1	1	yes	characteristic
469	C₂	—	4	2	no	—
470	C₂	—	16	2	no	—
471	C₂	C₆₇×#2856	1	1	yes	characteristic
472	C₂	C₆₇×#3057	1	2	yes	—
473	C₂	C₆₇×#3154	1	2	yes	—
474	C₂	C₆₇×#3174	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₆₇×#2619	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂×#117×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#492×C₆₇	C₂	semidirect
2. C₂×#468×C₆₇	C₂	semidirect
3. C₂×#465×C₆₇	C₂	semidirect
4. C₂×#405×C₆₇	C₂	semidirect
5. C₆₇×#2621	C₂	semidirect
7. C₆₇×#1973	C₂	semidirect
8. C₆₇×#1968	C₂	semidirect
9. C₆₇×#1961	C₂	semidirect
10. C₆₇×#1960	C₂	semidirect
11. C₆₇×#1846	C₂	semidirect
56. C₂×M₁₆×C₆₇	D₄	semidirect
59. C₂²×C₈×C₆₇	D₄	semidirect
94. C₄×C₈×C₆₇	D₄	semidirect
231. #33178	C₆₇	direct (N × H)
238. C₂×#492	C₂×C₆₇	semidirect
239. C₂×#468	C₂×C₆₇	semidirect
240. C₂×#465	C₂×C₆₇	semidirect
241. C₂×#405	C₂×C₆₇	semidirect
242. #2621	C₂×C₆₇	semidirect
244. #1973	C₂×C₆₇	semidirect
245. #1968	C₂×C₆₇	semidirect
246. #1961	C₂×C₆₇	semidirect
247. #1960	C₂×C₆₇	semidirect
248. #1846	C₂×C₆₇	semidirect
249. C₆₇	#33178	direct (N × H)
294. C₂×M₁₆	D₄×C₆₇	semidirect
297. C₂²×C₈	D₄×C₆₇	semidirect
332. C₄×C₈	D₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#492×C₆₇	2
C₂×#468×C₆₇	2
C₂×#465×C₆₇	2
C₂×#405×C₆₇	2
C₆₇×#2621	2
C₆₇×#2619	2
C₆₇×#1973	2
C₆₇×#1968	2
C₆₇×#1961	2
C₆₇×#1960	2
C₆₇×#1846	2
#33178	67

Subgroups — C67×#33178 #30712429

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33178 #30712429