Subgroups — C₆₇×#21399 #30700650

Order 17152.

1214 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#540×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#433×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2274	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#1989	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1982	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1875	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1874	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1762	C₂	1	2	yes	maximal, complemented
9	C₄×#102×C₆₇	C₂	1	2	yes	maximal, complemented
10	#540×C₆₇	—	2	4	no	—
11	#540×C₆₇	C₄	1	8	yes	complemented
12	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×#125×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×#123×C₆₇	—	2	2	no	—
15	C₂×#121×C₆₇	—	2	2	no	—
16	C₂×#121×C₆₇	C₂²	1	2	yes	—
17	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
18	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
20	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
21	#433×C₆₇	—	2	4	no	—
22	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
23	C₂×#102×C₆₇	C₂²	1	2	yes	—
24	C₂×#102×C₆₇	C₂²	1	2	yes	—
25	C₂×#102×C₆₇	C₂²	1	2	yes	—
26	C₂×#102×C₆₇	C₂²	1	2	yes	—
27	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
28	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
29	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
30	C₄×C₄⋊C₄×C₆₇	—	2	2	no	—
31	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
32	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
33	C₄×C₂²⋊C₄×C₆₇	—	2	2	no	—
34	#340×C₆₇	C₂²	1	2	yes	—
35	#340×C₆₇	C₂²	1	2	yes	—
36	#340×C₆₇	C₂²	1	2	yes	—
37	#340×C₆₇	C₂²	1	2	yes	—
38	C₂²×D₄×C₆₇	—	2	2	no	—
39	C₂²×D₄×C₆₇	—	2	2	no	—
40	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
41	C₂³×C₄×C₆₇	—	2	1	no	—
42	C₂³×C₄×C₆₇	—	2	2	no	—
43	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
44	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
45	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
46	#127×C₆₇	—	2	2	no	—
47	#127×C₆₇	C₂×C₄	1	4	yes	complemented
48	#125×C₆₇	—	2	2	no	—
49	#125×C₆₇	C₂×C₄	1	4	yes	complemented
50	#123×C₆₇	—	2	16	no	—
51	#121×C₆₇	—	2	4	no	—
52	#121×C₆₇	—	2	16	no	—
53	#121×C₆₇	C₂×C₄	1	8	yes	—
54	C₄×D₄×C₆₇	—	2	4	no	—
55	C₄×D₄×C₆₇	C₂×C₄	1	8	yes	complemented
56	#117×C₆₇	—	2	4	no	—
57	#117×C₆₇	C₂×C₄	1	4	yes	—
58	#117×C₆₇	D₄	1	4	yes	—
59	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
60	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
61	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
62	C₂×C₄⋊C₄×C₆₇	D₄	1	2	yes	—
63	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
64	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
65	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
66	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
67	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
68	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
69	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
70	C₂×C₄²×C₆₇	—	2	1	no	—
71	C₂×C₄²×C₆₇	—	2	2	no	—
72	C₂×C₄²×C₆₇	—	2	2	no	—
73	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
74	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
75	C₄⋊C₈×C₆₇	—	2	4	no	—
76	#102×C₆₇	—	2	4	no	—
77	#102×C₆₇	—	2	4	no	—
78	#102×C₆₇	—	2	4	no	—
79	#102×C₆₇	—	2	4	no	—
80	C₂²⋊C₈×C₆₇	—	2	4	no	—
81	#97×C₆₇	—	2	2	no	—
82	C₄×C₈×C₆₇	—	2	2	no	—
83	#95×C₆₇	—	2	2	no	—
84	#95×C₆₇	—	2	2	no	—
85	C₂⁴×C₆₇	—	2	2	no	—
86	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
87	C₂×D₄×C₆₇	—	2	8	no	—
88	C₂×D₄×C₆₇	—	2	8	no	—
89	C₂×D₄×C₆₇	—	2	2	no	—
90	C₂×D₄×C₆₇	—	2	2	no	—
91	C₂×D₄×C₆₇	—	2	2	no	—
92	C₂×D₄×C₆₇	—	2	2	no	—
93	C₂×D₄×C₆₇	—	2	2	no	—
94	C₂×D₄×C₆₇	—	8	2	no	—
95	C₂×D₄×C₆₇	—	8	2	no	—
96	C₂×D₄×C₆₇	—	8	2	no	—
97	C₂×D₄×C₆₇	—	8	2	no	—
98	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
99	C₂²×C₄×C₆₇	—	2	2	no	—
100	C₂²×C₄×C₆₇	—	2	2	no	—
101	C₂²×C₄×C₆₇	—	2	2	no	—
102	C₂²×C₄×C₆₇	—	2	2	no	—
103	C₂²×C₄×C₆₇	—	2	2	no	—
104	C₂²×C₄×C₆₇	—	2	4	no	—
105	C₂²×C₄×C₆₇	—	2	8	no	—
106	C₂²×C₄×C₆₇	—	8	2	no	—
107	C₂²×C₄×C₆₇	—	8	2	no	—
108	C₂²×C₄×C₆₇	—	8	2	no	—
109	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
110	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
111	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
112	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
113	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
114	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
115	C₂×C₈×C₆₇	—	2	2	no	—
116	C₂×C₈×C₆₇	—	2	2	no	—
117	C₂×C₈×C₆₇	—	2	2	no	—
118	C₂×C₈×C₆₇	—	2	2	no	—
119	C₂×C₈×C₆₇	—	2	2	no	—
120	C₂×C₈×C₆₇	—	2	2	no	—
121	C₄⋊C₄×C₆₇	—	2	4	no	—
122	C₄⋊C₄×C₆₇	—	2	8	no	—
123	C₄⋊C₄×C₆₇	—	2	2	no	—
124	C₄⋊C₄×C₆₇	—	2	2	no	—
125	C₄⋊C₄×C₆₇	—	2	2	no	—
126	C₄⋊C₄×C₆₇	C₂²⋊C₄	1	8	yes	—
127	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
128	C₄⋊C₄×C₆₇	#41	1	4	yes	—
129	C₄⋊C₄×C₆₇	#41	1	4	yes	—
130	C₂²⋊C₄×C₆₇	—	2	4	no	—
131	C₂²⋊C₄×C₆₇	—	2	8	no	—
132	C₂²⋊C₄×C₆₇	—	2	8	no	—
133	C₂²⋊C₄×C₆₇	C₂²⋊C₄	1	8	yes	—
134	C₄²×C₆₇	—	2	4	no	—
135	C₄²×C₆₇	—	8	2	no	—
136	C₄²×C₆₇	—	8	2	no	—
137	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
138	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
139	C₄²×C₆₇	C₂×D₄	1	2	yes	—
140	C₄²×C₆₇	C₂×D₄	1	2	yes	—
141	C₂³×C₆₇	—	2	1	no	—
142	C₂³×C₆₇	—	2	1	no	—
143	C₂³×C₆₇	—	2	2	no	—
144	C₂³×C₆₇	—	2	2	no	—
145	C₂³×C₆₇	—	2	4	no	—
146	C₂³×C₆₇	—	4	2	no	—
147	C₂³×C₆₇	—	4	2	no	—
148	C₂³×C₆₇	—	4	2	no	—
149	C₂³×C₆₇	—	4	2	no	—
150	C₂³×C₆₇	—	4	2	no	—
151	C₂³×C₆₇	—	4	2	no	—
152	C₂³×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
153	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
154	D₄×C₆₇	—	2	4	no	—
155	D₄×C₆₇	—	2	4	no	—
156	D₄×C₆₇	—	4	2	no	—
157	D₄×C₆₇	—	4	2	no	—
158	D₄×C₆₇	—	8	8	no	—
159	D₄×C₆₇	—	8	8	no	—
160	C₂×C₄×C₆₇	—	2	1	no	—
161	C₂×C₄×C₆₇	—	2	1	no	—
162	C₂×C₄×C₆₇	—	2	2	no	—
163	C₂×C₄×C₆₇	—	2	2	no	—
164	C₂×C₄×C₆₇	—	2	2	no	—
165	C₂×C₄×C₆₇	—	2	2	no	—
166	C₂×C₄×C₆₇	—	2	4	no	—
167	C₂×C₄×C₆₇	—	2	4	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	—	4	2	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	2	no	—
176	C₂×C₄×C₆₇	—	4	2	no	—
177	C₂×C₄×C₆₇	—	8	8	no	—
178	C₂×C₄×C₆₇	—	8	4	no	—
179	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
180	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
181	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
182	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
183	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
184	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
185	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
186	C₈×C₆₇	—	4	2	no	—
187	C₈×C₆₇	—	4	2	no	—
188	C₂²×C₆₇	—	2	4	no	—
189	C₂²×C₆₇	—	2	2	no	—
190	C₂²×C₆₇	—	4	2	no	—
191	C₂²×C₆₇	—	4	1	no	—
192	C₂²×C₆₇	—	4	1	no	—
193	C₂²×C₆₇	—	4	4	no	—
194	C₂²×C₆₇	—	4	4	no	—
195	C₂²×C₆₇	—	4	4	no	—
196	C₂²×C₆₇	—	8	2	no	—
197	C₂²×C₆₇	—	8	2	no	—
198	C₂²×C₆₇	—	8	2	no	—
199	C₂²×C₆₇	—	8	2	no	—
200	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
201	C₂²×C₆₇	#525	1	1	yes	characteristic
202	C₂²×C₆₇	#525	1	1	yes	characteristic
203	C₂²×C₆₇	#576	1	2	yes	—
204	C₂²×C₆₇	#577	1	2	yes	—
205	C₄×C₆₇	—	2	4	no	—
206	C₄×C₆₇	—	2	4	no	—
207	C₄×C₆₇	—	2	2	no	—
208	C₄×C₆₇	—	4	4	no	—
209	C₄×C₆₇	—	4	4	no	—
210	C₄×C₆₇	—	4	4	no	—
211	C₄×C₆₇	C₄×#41	1	2	yes	—
212	C₄×C₆₇	#532	1	2	yes	—
213	#21399	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
214	C₂×C₆₇	—	4	2	no	—
215	C₂×C₆₇	—	8	4	no	—
216	C₂×C₆₇	#2297	1	1	yes	characteristic
217	C₂×C₆₇	#2897	1	2	yes	—
218	C₂×C₆₇	#3119	1	2	yes	—
219	C₂×C₆₇	#3122	1	1	yes	characteristic
220	C₂×C₆₇	#3126	1	1	yes	characteristic
221	C₂×#540	C₂×C₆₇	1	1	yes	characteristic
222	C₂×#433	C₂×C₆₇	1	1	yes	characteristic, complemented
223	#2274	C₂×C₆₇	1	1	yes	characteristic, complemented
224	#1989	C₂×C₆₇	1	2	yes	complemented
225	#1982	C₂×C₆₇	1	2	yes	complemented
226	#1875	C₂×C₆₇	1	2	yes	complemented
227	#1874	C₂×C₆₇	1	2	yes	complemented
228	#1762	C₂×C₆₇	1	2	yes	complemented
229	C₄×#102	C₂×C₆₇	1	2	yes	complemented
230	C₆₇	#21399	1	1	yes	characteristic, direct product factor, Sylow
231	#540	—	2	4	no	—
232	#540	C₄×C₆₇	1	8	yes	complemented
233	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
234	C₂×#125	C₂²×C₆₇	1	1	yes	characteristic
235	C₂×#123	—	2	2	no	—
236	C₂×#121	—	2	2	no	—
237	C₂×#121	C₂²×C₆₇	1	2	yes	—
238	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
239	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
240	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
241	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
242	#433	—	2	4	no	—
243	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
244	C₂×#102	C₂²×C₆₇	1	2	yes	—
245	C₂×#102	C₂²×C₆₇	1	2	yes	—
246	C₂×#102	C₂²×C₆₇	1	2	yes	—
247	C₂×#102	C₂²×C₆₇	1	2	yes	—
248	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
249	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
250	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
251	C₄×C₄⋊C₄	—	2	2	no	—
252	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
253	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
254	C₄×C₂²⋊C₄	—	2	2	no	—
255	#340	C₂²×C₆₇	1	2	yes	—
256	#340	C₂²×C₆₇	1	2	yes	—
257	#340	C₂²×C₆₇	1	2	yes	—
258	#340	C₂²×C₆₇	1	2	yes	—
259	C₂²×D₄	—	2	2	no	—
260	C₂²×D₄	—	2	2	no	—
261	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
262	C₂³×C₄	—	2	1	no	—
263	C₂³×C₄	—	2	2	no	—
264	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
265	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
266	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
267	#127	—	2	2	no	—
268	#127	C₂×C₄×C₆₇	1	4	yes	complemented
269	#125	—	2	2	no	—
270	#125	C₂×C₄×C₆₇	1	4	yes	complemented
271	#123	—	2	16	no	—
272	#121	—	2	4	no	—
273	#121	—	2	16	no	—
274	#121	C₂×C₄×C₆₇	1	8	yes	—
275	C₄×D₄	—	2	4	no	—
276	C₄×D₄	C₂×C₄×C₆₇	1	8	yes	complemented
277	#117	—	2	4	no	—
278	#117	C₂×C₄×C₆₇	1	4	yes	—
279	#117	D₄×C₆₇	1	4	yes	—
280	C₂×C₄⋊C₄	—	2	1	no	—
281	C₂×C₄⋊C₄	—	2	2	no	—
282	C₂×C₄⋊C₄	—	2	4	no	—
283	C₂×C₄⋊C₄	D₄×C₆₇	1	2	yes	—
284	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
285	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
286	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
287	C₂×C₂²⋊C₄	—	2	1	no	—
288	C₂×C₂²⋊C₄	—	2	2	no	—
289	C₂×C₂²⋊C₄	—	2	2	no	—
290	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
291	C₂×C₄²	—	2	1	no	—
292	C₂×C₄²	—	2	2	no	—
293	C₂×C₄²	—	2	2	no	—
294	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
295	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
296	C₄⋊C₈	—	2	4	no	—
297	#102	—	2	4	no	—
298	#102	—	2	4	no	—
299	#102	—	2	4	no	—
300	#102	—	2	4	no	—
301	C₂²⋊C₈	—	2	4	no	—
302	#97	—	2	2	no	—
303	C₄×C₈	—	2	2	no	—
304	#95	—	2	2	no	—
305	#95	—	2	2	no	—
306	C₂⁴	—	2	2	no	—
307	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
308	C₂×D₄	—	2	8	no	—
309	C₂×D₄	—	2	8	no	—
310	C₂×D₄	—	2	2	no	—
311	C₂×D₄	—	2	2	no	—
312	C₂×D₄	—	2	2	no	—
313	C₂×D₄	—	2	2	no	—
314	C₂×D₄	—	2	2	no	—
315	C₂×D₄	—	8	2	no	—
316	C₂×D₄	—	8	2	no	—
317	C₂×D₄	—	8	2	no	—
318	C₂×D₄	—	8	2	no	—
319	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
320	C₂²×C₄	—	2	2	no	—
321	C₂²×C₄	—	2	2	no	—
322	C₂²×C₄	—	2	2	no	—
323	C₂²×C₄	—	2	2	no	—
324	C₂²×C₄	—	2	2	no	—
325	C₂²×C₄	—	2	4	no	—
326	C₂²×C₄	—	2	8	no	—
327	C₂²×C₄	—	8	2	no	—
328	C₂²×C₄	—	8	2	no	—
329	C₂²×C₄	—	8	2	no	—
330	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
331	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
332	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
333	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
334	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
335	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
336	C₂×C₈	—	2	2	no	—
337	C₂×C₈	—	2	2	no	—
338	C₂×C₈	—	2	2	no	—
339	C₂×C₈	—	2	2	no	—
340	C₂×C₈	—	2	2	no	—
341	C₂×C₈	—	2	2	no	—
342	C₄⋊C₄	—	2	4	no	—
343	C₄⋊C₄	—	2	8	no	—
344	C₄⋊C₄	—	2	2	no	—
345	C₄⋊C₄	—	2	2	no	—
346	C₄⋊C₄	—	2	2	no	—
347	C₄⋊C₄	C₂²⋊C₄×C₆₇	1	8	yes	—
348	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
349	C₄⋊C₄	#41×C₆₇	1	4	yes	—
350	C₄⋊C₄	#41×C₆₇	1	4	yes	—
351	C₂²⋊C₄	—	2	4	no	—
352	C₂²⋊C₄	—	2	8	no	—
353	C₂²⋊C₄	—	2	8	no	—
354	C₂²⋊C₄	C₂²⋊C₄×C₆₇	1	8	yes	—
355	C₄²	—	2	4	no	—
356	C₄²	—	8	2	no	—
357	C₄²	—	8	2	no	—
358	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
359	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
360	C₄²	C₂×D₄×C₆₇	1	2	yes	—
361	C₄²	C₂×D₄×C₆₇	1	2	yes	—
362	C₂³	—	2	1	no	—
363	C₂³	—	2	1	no	—
364	C₂³	—	2	2	no	—
365	C₂³	—	2	2	no	—
366	C₂³	—	2	4	no	—
367	C₂³	—	4	2	no	—
368	C₂³	—	4	2	no	—
369	C₂³	—	4	2	no	—
370	C₂³	—	4	2	no	—
371	C₂³	—	4	2	no	—
372	C₂³	—	4	2	no	—
373	C₂³	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
374	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
375	D₄	—	2	4	no	—
376	D₄	—	2	4	no	—
377	D₄	—	4	2	no	—
378	D₄	—	4	2	no	—
379	D₄	—	8	8	no	—
380	D₄	—	8	8	no	—
381	C₂×C₄	—	2	1	no	—
382	C₂×C₄	—	2	1	no	—
383	C₂×C₄	—	2	2	no	—
384	C₂×C₄	—	2	2	no	—
385	C₂×C₄	—	2	2	no	—
386	C₂×C₄	—	2	2	no	—
387	C₂×C₄	—	2	4	no	—
388	C₂×C₄	—	2	4	no	—
389	C₂×C₄	—	2	4	no	—
390	C₂×C₄	—	2	4	no	—
391	C₂×C₄	—	2	4	no	—
392	C₂×C₄	—	4	2	no	—
393	C₂×C₄	—	4	2	no	—
394	C₂×C₄	—	4	2	no	—
395	C₂×C₄	—	4	2	no	—
396	C₂×C₄	—	4	2	no	—
397	C₂×C₄	—	4	2	no	—
398	C₂×C₄	—	8	8	no	—
399	C₂×C₄	—	8	4	no	—
400	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
401	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
402	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
403	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
404	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
405	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
406	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
407	C₈	—	4	2	no	—
408	C₈	—	4	2	no	—
409	C₂²	—	2	2	no	—
410	C₂²	—	2	4	no	—
411	C₂²	—	4	2	no	—
412	C₂²	—	4	1	no	—
413	C₂²	—	4	1	no	—
414	C₂²	—	4	4	no	—
415	C₂²	—	4	4	no	—
416	C₂²	—	4	4	no	—
417	C₂²	—	8	2	no	—
418	C₂²	—	8	2	no	—
419	C₂²	—	8	2	no	—
420	C₂²	—	8	2	no	—
421	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
422	C₂²	#525×C₆₇	1	1	yes	characteristic
423	C₂²	#525×C₆₇	1	1	yes	characteristic
424	C₂²	#576×C₆₇	1	2	yes	—
425	C₂²	#577×C₆₇	1	2	yes	—
426	C₄	—	2	2	no	—
427	C₄	—	2	4	no	—
428	C₄	—	2	4	no	—
429	C₄	—	4	4	no	—
430	C₄	—	4	4	no	—
431	C₄	—	4	4	no	—
432	C₄	C₄×#41×C₆₇	1	2	yes	—
433	C₄	#532×C₆₇	1	2	yes	—
434	C₂	—	4	2	no	—
435	C₂	—	8	4	no	—
436	C₂	C₆₇×#2297	1	1	yes	characteristic
437	C₂	C₆₇×#2897	1	2	yes	—
438	C₂	C₆₇×#3119	1	2	yes	—
439	C₂	C₆₇×#3122	1	1	yes	characteristic
440	C₂	C₆₇×#3126	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂×#540×C₆₇	C₂
LCS	G ▷ C₂×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
2. C₂×#433×C₆₇	C₂	semidirect
3. C₆₇×#2274	C₂	semidirect
4. C₆₇×#1989	C₂	semidirect
5. C₆₇×#1982	C₂	semidirect
6. C₆₇×#1875	C₂	semidirect
7. C₆₇×#1874	C₂	semidirect
8. C₆₇×#1762	C₂	semidirect
9. C₄×#102×C₆₇	C₂	semidirect
11. #540×C₆₇	C₄	semidirect
47. #127×C₆₇	C₂×C₄	semidirect
49. #125×C₆₇	C₂×C₄	semidirect
55. C₄×D₄×C₆₇	C₂×C₄	semidirect
213. #21399	C₆₇	direct (N × H)
222. C₂×#433	C₂×C₆₇	semidirect
223. #2274	C₂×C₆₇	semidirect
224. #1989	C₂×C₆₇	semidirect
225. #1982	C₂×C₆₇	semidirect
226. #1875	C₂×C₆₇	semidirect
227. #1874	C₂×C₆₇	semidirect
228. #1762	C₂×C₆₇	semidirect
229. C₄×#102	C₂×C₆₇	semidirect
230. C₆₇	#21399	direct (N × H)
232. #540	C₄×C₆₇	semidirect
268. #127	C₂×C₄×C₆₇	semidirect
270. #125	C₂×C₄×C₆₇	semidirect
276. C₄×D₄	C₂×C₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#540×C₆₇	2
C₂×#433×C₆₇	2
C₆₇×#2274	2
C₆₇×#1989	2
C₆₇×#1982	2
C₆₇×#1875	2
C₆₇×#1874	2
C₆₇×#1762	2
C₄×#102×C₆₇	2
#21399	67

Subgroups — C67×#21399 #30700650

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#21399 #30700650