Subgroups — C₆₇×#21252 #30700503

Order 17152.

1610 conjugacy classes of subgroups.

The subgroup lattice has 456 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
2	C₂×#421×C₆₇	C₂	1	4	yes	maximal, complemented
3	C₂×C₄×M₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#2285	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1980	C₂	1	8	yes	maximal, complemented
6	C₂²×#41×C₆₇	C₂²	1	2	yes	—
7	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
8	#532×C₆₇	—	2	1	no	—
9	#532×C₆₇	—	2	1	no	—
10	#532×C₆₇	—	2	1	no	—
11	#532×C₆₇	—	2	1	no	—
12	#532×C₆₇	C₄	1	4	yes	complemented
13	#532×C₆₇	C₂²	1	4	yes	—
14	C₂×C₄⋊Q₈×C₆₇	C₄	1	1	yes	characteristic, complemented
15	C₂×#127×C₆₇	C₄	1	1	yes	characteristic, complemented
16	C₂×#124×C₆₇	C₄	1	2	yes	complemented
17	C₂×#121×C₆₇	—	2	2	no	—
18	C₂×C₄×D₄×C₆₇	—	2	2	no	—
19	C₂×#117×C₆₇	C₂²	1	2	yes	—
20	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
21	#421×C₆₇	—	2	16	no	—
22	C₂×#104×C₆₇	C₂²	1	16	yes	complemented
23	C₄×M₁₆×C₆₇	—	2	2	no	—
24	C₄×M₁₆×C₆₇	C₂²	1	4	yes	complemented
25	C₂×#97×C₆₇	—	2	1	no	—
26	C₂×C₄×C₈×C₆₇	—	2	1	no	—
27	#384×C₆₇	—	2	1	no	—
28	#382×C₆₇	—	2	1	no	—
29	#380×C₆₇	—	2	2	no	—
30	C₄×C₄⋊C₄×C₆₇	C₂²	1	4	yes	complemented
31	C₂×#41×C₆₇	—	2	2	no	—
32	C₂×#41×C₆₇	—	2	2	no	—
33	C₂×#41×C₆₇	—	2	8	no	—
34	C₂×#41×C₆₇	—	2	8	no	—
35	C₂×#41×C₆₇	C₂×C₄	1	4	yes	—
36	C₂×#41×C₆₇	C₂³	1	4	yes	—
37	C₂²×Q₈×C₆₇	C₂×C₄	1	2	yes	—
38	C₂²×D₄×C₆₇	—	2	2	no	—
39	C₂²×D₄×C₆₇	—	2	2	no	—
40	C₂²×D₄×C₆₇	C₂×C₄	1	2	yes	—
41	C₂³×C₄×C₆₇	—	2	2	no	—
42	C₂³×C₄×C₆₇	D₄	1	2	yes	—
43	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
44	C₂×M₁₆×C₆₇	—	2	2	no	—
45	C₂×M₁₆×C₆₇	—	2	4	no	—
46	C₂×M₁₆×C₆₇	D₄	1	8	yes	complemented
47	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
48	C₂²×C₈×C₆₇	—	2	2	no	—
49	C₄⋊Q₈×C₆₇	—	2	1	no	—
50	C₄⋊Q₈×C₆₇	—	2	1	no	—
51	C₄⋊Q₈×C₆₇	C₂×C₄	1	4	yes	—
52	#127×C₆₇	—	2	1	no	—
53	#127×C₆₇	—	2	1	no	—
54	#127×C₆₇	C₂×C₄	1	4	yes	—
55	#124×C₆₇	—	2	2	no	—
56	#124×C₆₇	—	2	2	no	—
57	#124×C₆₇	C₂×C₄	1	8	yes	—
58	#121×C₆₇	—	2	4	no	—
59	#121×C₆₇	—	2	4	no	—
60	#121×C₆₇	—	2	4	no	—
61	#121×C₆₇	—	2	4	no	—
62	C₄×D₄×C₆₇	—	2	4	no	—
63	C₄×D₄×C₆₇	—	2	4	no	—
64	C₄×D₄×C₆₇	—	2	4	no	—
65	C₄×D₄×C₆₇	—	2	4	no	—
66	#117×C₆₇	—	2	8	no	—
67	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
68	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
69	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
70	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
71	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
72	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
73	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
74	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
75	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
76	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
77	C₂×C₄²×C₆₇	D₄	1	2	yes	—
78	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
79	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
80	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
81	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
82	#104×C₆₇	—	2	32	no	—
83	#97×C₆₇	—	2	4	no	—
84	C₄×C₈×C₆₇	—	2	4	no	—
85	#95×C₆₇	—	2	4	no	—
86	C₂⁴×C₆₇	—	2	2	no	—
87	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
88	#41×C₆₇	—	2	16	no	—
89	#41×C₆₇	—	2	8	no	—
90	#41×C₆₇	—	2	8	no	—
91	#41×C₆₇	—	4	8	no	—
92	#41×C₆₇	—	4	4	no	—
93	#41×C₆₇	—	4	4	no	—
94	C₂×Q₈×C₆₇	—	2	2	no	—
95	C₂×Q₈×C₆₇	—	2	8	no	—
96	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
97	C₂×D₄×C₆₇	—	2	4	no	—
98	C₂×D₄×C₆₇	—	2	4	no	—
99	C₂×D₄×C₆₇	—	2	4	no	—
100	C₂×D₄×C₆₇	—	2	4	no	—
101	C₂×D₄×C₆₇	—	2	2	no	—
102	C₂×D₄×C₆₇	—	2	8	no	—
103	C₂×D₄×C₆₇	—	4	8	no	—
104	C₂×D₄×C₆₇	—	4	4	no	—
105	C₂×D₄×C₆₇	—	4	4	no	—
106	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
107	C₂²×C₄×C₆₇	—	2	2	no	—
108	C₂²×C₄×C₆₇	—	2	4	no	—
109	C₂²×C₄×C₆₇	—	2	4	no	—
110	C₂²×C₄×C₆₇	—	2	4	no	—
111	C₂²×C₄×C₆₇	—	4	4	no	—
112	C₂²×C₄×C₆₇	—	4	8	no	—
113	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
114	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
115	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
116	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
117	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
118	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
119	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
120	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
121	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
122	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
123	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
124	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
125	C₂²×C₄×C₆₇	#41	1	4	yes	—
126	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
127	M₁₆×C₆₇	—	2	8	no	—
128	M₁₆×C₆₇	—	2	4	no	—
129	M₁₆×C₆₇	C₂×D₄	1	8	yes	complemented
130	C₂×C₈×C₆₇	—	2	8	no	—
131	C₂×C₈×C₆₇	—	2	4	no	—
132	C₄⋊C₄×C₆₇	—	2	4	no	—
133	C₄⋊C₄×C₆₇	—	2	4	no	—
134	C₄⋊C₄×C₆₇	—	4	8	no	—
135	C₄⋊C₄×C₆₇	—	4	4	no	—
136	C₂²⋊C₄×C₆₇	—	2	8	no	—
137	C₂²⋊C₄×C₆₇	—	2	8	no	—
138	C₂²⋊C₄×C₆₇	—	4	4	no	—
139	C₄²×C₆₇	—	2	8	no	—
140	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
141	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
142	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
143	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
144	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
145	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
146	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
147	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
148	C₄²×C₆₇	C₂×D₄	1	2	yes	—
149	C₄²×C₆₇	C₂×D₄	1	2	yes	—
150	C₂³×C₆₇	—	2	4	no	—
151	C₂³×C₆₇	—	4	2	no	—
152	C₂³×C₆₇	—	4	8	no	—
153	C₂³×C₆₇	—	4	4	no	—
154	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
155	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
156	C₂³×C₆₇	#124	1	2	yes	—
157	C₂³×C₆₇	#127	1	2	yes	—
158	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
159	Q₈×C₆₇	—	2	8	no	—
160	Q₈×C₆₇	—	4	4	no	—
161	D₄×C₆₇	—	2	8	no	—
162	D₄×C₆₇	—	4	8	no	—
163	D₄×C₆₇	—	4	4	no	—
164	D₄×C₆₇	—	4	4	no	—
165	D₄×C₆₇	—	4	8	no	—
166	D₄×C₆₇	—	4	4	no	—
167	D₄×C₆₇	—	4	8	no	—
168	C₂×C₄×C₆₇	—	2	4	no	—
169	C₂×C₄×C₆₇	—	2	4	no	—
170	C₂×C₄×C₆₇	—	2	4	no	—
171	C₂×C₄×C₆₇	—	2	4	no	—
172	C₂×C₄×C₆₇	—	4	2	no	—
173	C₂×C₄×C₆₇	—	4	2	no	—
174	C₂×C₄×C₆₇	—	4	2	no	—
175	C₂×C₄×C₆₇	—	4	4	no	—
176	C₂×C₄×C₆₇	—	4	8	no	—
177	C₂×C₄×C₆₇	—	4	8	no	—
178	C₂×C₄×C₆₇	—	4	8	no	—
179	C₂×C₄×C₆₇	—	4	4	no	—
180	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
181	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
182	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
183	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
184	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
185	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
186	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
187	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
188	C₂×C₄×C₆₇	#121	1	4	yes	—
189	C₂×C₄×C₆₇	#121	1	4	yes	—
190	C₂×C₄×C₆₇	#124	1	2	yes	—
191	C₂×C₄×C₆₇	#127	1	2	yes	—
192	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
193	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
194	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
195	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
196	C₈×C₆₇	—	2	8	no	—
197	C₂²×C₆₇	—	2	2	no	—
198	C₂²×C₆₇	—	2	2	no	—
199	C₂²×C₆₇	—	2	2	no	—
200	C₂²×C₆₇	—	4	2	no	—
201	C₂²×C₆₇	—	4	2	no	—
202	C₂²×C₆₇	—	4	8	no	—
203	C₂²×C₆₇	—	4	4	no	—
204	C₂²×C₆₇	—	8	8	no	—
205	C₂²×C₆₇	#390	1	2	yes	—
206	C₂²×C₆₇	#390	1	2	yes	—
207	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
208	C₂²×C₆₇	C₂×#124	1	1	yes	characteristic
209	C₂²×C₆₇	C₂×#127	1	1	yes	characteristic
210	C₄×C₆₇	—	2	2	no	—
211	C₄×C₆₇	—	2	2	no	—
212	C₄×C₆₇	—	2	2	no	—
213	C₄×C₆₇	—	4	4	no	—
214	C₄×C₆₇	—	8	4	no	—
215	C₄×C₆₇	#390	1	2	yes	—
216	C₄×C₆₇	#390	1	2	yes	—
217	C₄×C₆₇	#421	1	4	yes	complemented
218	C₄×C₆₇	C₂×C₄×D₄	1	2	yes	—
219	C₄×C₆₇	C₂×#121	1	2	yes	—
220	#21252	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
221	C₂×C₆₇	—	2	1	no	—
222	C₂×C₆₇	—	2	2	no	—
223	C₂×C₆₇	—	2	1	no	—
224	C₂×C₆₇	—	2	2	no	—
225	C₂×C₆₇	—	8	4	no	—
226	C₂×C₆₇	C₂×#390	1	1	yes	characteristic
227	C₂×C₆₇	C₂×#421	1	2	yes	—
228	C₂×#532	C₂×C₆₇	1	1	yes	characteristic
229	C₂×#421	C₂×C₆₇	1	4	yes	complemented
230	C₂×C₄×M₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
231	#2285	C₂×C₆₇	1	1	yes	characteristic, complemented
232	#1980	C₂×C₆₇	1	8	yes	complemented
233	C₆₇	#21252	1	1	yes	characteristic, direct product factor, Sylow
234	C₂²×#41	C₂²×C₆₇	1	2	yes	—
235	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
236	#532	—	2	1	no	—
237	#532	—	2	1	no	—
238	#532	—	2	1	no	—
239	#532	—	2	1	no	—
240	#532	C₄×C₆₇	1	4	yes	complemented
241	#532	C₂²×C₆₇	1	4	yes	—
242	C₂×C₄⋊Q₈	C₄×C₆₇	1	1	yes	characteristic, complemented
243	C₂×#127	C₄×C₆₇	1	1	yes	characteristic, complemented
244	C₂×#124	C₄×C₆₇	1	2	yes	complemented
245	C₂×#121	—	2	2	no	—
246	C₂×C₄×D₄	—	2	2	no	—
247	C₂×#117	C₂²×C₆₇	1	2	yes	—
248	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
249	#421	—	2	16	no	—
250	C₂×#104	C₂²×C₆₇	1	16	yes	complemented
251	C₄×M₁₆	—	2	2	no	—
252	C₄×M₁₆	C₂²×C₆₇	1	4	yes	complemented
253	C₂×#97	—	2	1	no	—
254	C₂×C₄×C₈	—	2	1	no	—
255	#384	—	2	1	no	—
256	#382	—	2	1	no	—
257	#380	—	2	2	no	—
258	C₄×C₄⋊C₄	C₂²×C₆₇	1	4	yes	complemented
259	C₂×#41	—	2	2	no	—
260	C₂×#41	—	2	2	no	—
261	C₂×#41	—	2	8	no	—
262	C₂×#41	—	2	8	no	—
263	C₂×#41	C₂×C₄×C₆₇	1	4	yes	—
264	C₂×#41	C₂³×C₆₇	1	4	yes	—
265	C₂²×Q₈	C₂×C₄×C₆₇	1	2	yes	—
266	C₂²×D₄	—	2	2	no	—
267	C₂²×D₄	—	2	2	no	—
268	C₂²×D₄	C₂×C₄×C₆₇	1	2	yes	—
269	C₂³×C₄	—	2	2	no	—
270	C₂³×C₄	D₄×C₆₇	1	2	yes	—
271	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
272	C₂×M₁₆	—	2	2	no	—
273	C₂×M₁₆	—	2	4	no	—
274	C₂×M₁₆	D₄×C₆₇	1	8	yes	complemented
275	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
276	C₂²×C₈	—	2	2	no	—
277	C₄⋊Q₈	—	2	1	no	—
278	C₄⋊Q₈	—	2	1	no	—
279	C₄⋊Q₈	C₂×C₄×C₆₇	1	4	yes	—
280	#127	—	2	1	no	—
281	#127	—	2	1	no	—
282	#127	C₂×C₄×C₆₇	1	4	yes	—
283	#124	—	2	2	no	—
284	#124	—	2	2	no	—
285	#124	C₂×C₄×C₆₇	1	8	yes	—
286	#121	—	2	4	no	—
287	#121	—	2	4	no	—
288	#121	—	2	4	no	—
289	#121	—	2	4	no	—
290	C₄×D₄	—	2	4	no	—
291	C₄×D₄	—	2	4	no	—
292	C₄×D₄	—	2	4	no	—
293	C₄×D₄	—	2	4	no	—
294	#117	—	2	8	no	—
295	C₂×C₄⋊C₄	—	2	2	no	—
296	C₂×C₄⋊C₄	—	2	2	no	—
297	C₂×C₄⋊C₄	—	2	4	no	—
298	C₂×C₂²⋊C₄	—	2	4	no	—
299	C₂×C₂²⋊C₄	—	2	2	no	—
300	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
301	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
302	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
303	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
304	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
305	C₂×C₄²	D₄×C₆₇	1	2	yes	—
306	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
307	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
308	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
309	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
310	#104	—	2	32	no	—
311	#97	—	2	4	no	—
312	C₄×C₈	—	2	4	no	—
313	#95	—	2	4	no	—
314	C₂⁴	—	2	2	no	—
315	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
316	#41	—	2	16	no	—
317	#41	—	2	8	no	—
318	#41	—	2	8	no	—
319	#41	—	4	8	no	—
320	#41	—	4	4	no	—
321	#41	—	4	4	no	—
322	C₂×Q₈	—	2	2	no	—
323	C₂×Q₈	—	2	8	no	—
324	C₂×Q₈	C₂²×C₄×C₆₇	1	4	yes	—
325	C₂×D₄	—	2	4	no	—
326	C₂×D₄	—	2	4	no	—
327	C₂×D₄	—	2	4	no	—
328	C₂×D₄	—	2	4	no	—
329	C₂×D₄	—	2	2	no	—
330	C₂×D₄	—	2	8	no	—
331	C₂×D₄	—	4	8	no	—
332	C₂×D₄	—	4	4	no	—
333	C₂×D₄	—	4	4	no	—
334	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
335	C₂²×C₄	—	2	2	no	—
336	C₂²×C₄	—	2	4	no	—
337	C₂²×C₄	—	2	4	no	—
338	C₂²×C₄	—	2	4	no	—
339	C₂²×C₄	—	4	4	no	—
340	C₂²×C₄	—	4	8	no	—
341	C₂²×C₄	C₂²⋊C₄×C₆₇	1	2	yes	—
342	C₂²×C₄	C₂²⋊C₄×C₆₇	1	2	yes	—
343	C₂²×C₄	C₂²⋊C₄×C₆₇	1	2	yes	—
344	C₂²×C₄	C₂²⋊C₄×C₆₇	1	2	yes	—
345	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
346	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
347	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
348	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
349	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
350	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
351	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
352	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
353	C₂²×C₄	#41×C₆₇	1	4	yes	—
354	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
355	M₁₆	—	2	8	no	—
356	M₁₆	—	2	4	no	—
357	M₁₆	C₂×D₄×C₆₇	1	8	yes	complemented
358	C₂×C₈	—	2	8	no	—
359	C₂×C₈	—	2	4	no	—
360	C₄⋊C₄	—	2	4	no	—
361	C₄⋊C₄	—	2	4	no	—
362	C₄⋊C₄	—	4	8	no	—
363	C₄⋊C₄	—	4	4	no	—
364	C₂²⋊C₄	—	2	8	no	—
365	C₂²⋊C₄	—	2	8	no	—
366	C₂²⋊C₄	—	4	4	no	—
367	C₄²	—	2	8	no	—
368	C₄²	C₂²⋊C₄×C₆₇	1	2	yes	—
369	C₄²	C₂²⋊C₄×C₆₇	1	2	yes	—
370	C₄²	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
371	C₄²	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
372	C₄²	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
373	C₄²	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
374	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
375	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
376	C₄²	C₂×D₄×C₆₇	1	2	yes	—
377	C₄²	C₂×D₄×C₆₇	1	2	yes	—
378	C₂³	—	2	4	no	—
379	C₂³	—	4	2	no	—
380	C₂³	—	4	8	no	—
381	C₂³	—	4	4	no	—
382	C₂³	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
383	C₂³	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
384	C₂³	#124×C₆₇	1	2	yes	—
385	C₂³	#127×C₆₇	1	2	yes	—
386	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
387	Q₈	—	2	8	no	—
388	Q₈	—	4	4	no	—
389	D₄	—	2	8	no	—
390	D₄	—	4	8	no	—
391	D₄	—	4	4	no	—
392	D₄	—	4	4	no	—
393	D₄	—	4	8	no	—
394	D₄	—	4	4	no	—
395	D₄	—	4	8	no	—
396	C₂×C₄	—	2	4	no	—
397	C₂×C₄	—	2	4	no	—
398	C₂×C₄	—	2	4	no	—
399	C₂×C₄	—	2	4	no	—
400	C₂×C₄	—	4	2	no	—
401	C₂×C₄	—	4	2	no	—
402	C₂×C₄	—	4	2	no	—
403	C₂×C₄	—	4	4	no	—
404	C₂×C₄	—	4	8	no	—
405	C₂×C₄	—	4	8	no	—
406	C₂×C₄	—	4	8	no	—
407	C₂×C₄	—	4	4	no	—
408	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
409	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
410	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
411	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
412	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
413	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
414	C₂×C₄	C₄×D₄×C₆₇	1	4	yes	—
415	C₂×C₄	C₄×D₄×C₆₇	1	4	yes	—
416	C₂×C₄	#121×C₆₇	1	4	yes	—
417	C₂×C₄	#121×C₆₇	1	4	yes	—
418	C₂×C₄	#124×C₆₇	1	2	yes	—
419	C₂×C₄	#127×C₆₇	1	2	yes	—
420	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
421	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
422	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
423	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
424	C₈	—	2	8	no	—
425	C₂²	—	2	2	no	—
426	C₂²	—	2	2	no	—
427	C₂²	—	2	2	no	—
428	C₂²	—	4	2	no	—
429	C₂²	—	4	2	no	—
430	C₂²	—	4	4	no	—
431	C₂²	—	4	8	no	—
432	C₂²	—	8	8	no	—
433	C₂²	#390×C₆₇	1	2	yes	—
434	C₂²	#390×C₆₇	1	2	yes	—
435	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
436	C₂²	C₂×#124×C₆₇	1	1	yes	characteristic
437	C₂²	C₂×#127×C₆₇	1	1	yes	characteristic
438	C₄	—	2	2	no	—
439	C₄	—	2	2	no	—
440	C₄	—	2	2	no	—
441	C₄	—	4	4	no	—
442	C₄	—	8	4	no	—
443	C₄	#390×C₆₇	1	2	yes	—
444	C₄	#390×C₆₇	1	2	yes	—
445	C₄	#421×C₆₇	1	4	yes	complemented
446	C₄	C₂×C₄×D₄×C₆₇	1	2	yes	—
447	C₄	C₂×#121×C₆₇	1	2	yes	—
448	C₂	—	2	1	no	—
449	C₂	—	2	2	no	—
450	C₂	—	2	1	no	—
451	C₂	—	2	2	no	—
452	C₂	—	8	4	no	—
453	C₂	C₂×#390×C₆₇	1	1	yes	characteristic
454	C₂	C₂×#421×C₆₇	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₆₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
socle	C₂²×C₆₇	C₂²×C₂²⋊C₄
prime radical	C₂×#532×C₆₇	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ C₂²×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#421×C₆₇	C₂	semidirect
3. C₂×C₄×M₁₆×C₆₇	C₂	semidirect
4. C₆₇×#2285	C₂	semidirect
5. C₆₇×#1980	C₂	semidirect
12. #532×C₆₇	C₄	semidirect
14. C₂×C₄⋊Q₈×C₆₇	C₄	semidirect
15. C₂×#127×C₆₇	C₄	semidirect
16. C₂×#124×C₆₇	C₄	semidirect
22. C₂×#104×C₆₇	C₂²	semidirect
24. C₄×M₁₆×C₆₇	C₂²	semidirect
30. C₄×C₄⋊C₄×C₆₇	C₂²	semidirect
46. C₂×M₁₆×C₆₇	D₄	semidirect
129. M₁₆×C₆₇	C₂×D₄	semidirect
217. C₄×C₆₇	#421	semidirect
220. #21252	C₆₇	direct (N × H)
229. C₂×#421	C₂×C₆₇	semidirect
230. C₂×C₄×M₁₆	C₂×C₆₇	semidirect
231. #2285	C₂×C₆₇	semidirect
232. #1980	C₂×C₆₇	semidirect
233. C₆₇	#21252	direct (N × H)
240. #532	C₄×C₆₇	semidirect
242. C₂×C₄⋊Q₈	C₄×C₆₇	semidirect
243. C₂×#127	C₄×C₆₇	semidirect
244. C₂×#124	C₄×C₆₇	semidirect
250. C₂×#104	C₂²×C₆₇	semidirect
252. C₄×M₁₆	C₂²×C₆₇	semidirect
258. C₄×C₄⋊C₄	C₂²×C₆₇	semidirect
274. C₂×M₁₆	D₄×C₆₇	semidirect
357. M₁₆	C₂×D₄×C₆₇	semidirect
445. C₄	#421×C₆₇	semidirect

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
C₂×#421×C₆₇	2
C₂×C₄×M₁₆×C₆₇	2
C₆₇×#2285	2
C₆₇×#1980	2
#21252	67

Subgroups — C67×#21252 #30700503

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#21252 #30700503